Generation of entangled qubit states

ABSTRACT

A method includes receiving Bell pairs. Photons are obtained in a Greenberger-Horn-Zeilinger (GHZ) state by providing, to a first beam splitter, a photon from a first Bell pair and a photon from a second Bell pair. The first beam splitter is coupled with a first output channel and a second output channel. Obtaining the photons in the GHZ state further includes providing, to a second beam splitter, a photon from a third Bell pair and a photon from a fourth Bell pair. The second beam splitter is coupled with a third output channel and a fourth output channel. Obtaining the photons in the GHZ state further includes providing a photon output from the second output channel as a first input to a detector and a photon output in the third output channel a second input to the first detector.

RELATED APPLICATIONS

This application is a continuation application of U.S. patentapplication Ser. No. 16/621,994 filed on Dec. 12, 2019, which is a U.S.National Stage Application filed under 35 U.S.C. § 371 of PCT PatentApplication Serial No. PCT/US2019/023756 filed on Mar. 22, 2019, whichclaims the benefit of and priority to U.S. Provisional Application No.62/647,557, filed Mar. 23, 2018, entitled “Generation Of EntangledPhotonic States”; U.S. Provisional Application No. 62/715,607, filedAug. 7, 2018, entitled “Generation Of Entangled Photonic States”; U.S.Provisional Application No. 62/770,645, filed Nov. 21, 2018, entitled“Generation Of Entangled Photonic States”; and U.S. ProvisionalApplication No. 62/770,648, filed Nov. 21, 2018, entitled “Generation ofa Cluster State for Universal Quantum Computing from Bell Pairs”, eachof which is incorporated by reference herein in its entirety.

TECHNICAL FIELD

This relates generally to quantum technology devices (e.g., hybridelectronic/photonic devices) and, more specifically, to quantumtechnology devices for generating entangled states of qubits (e.g.,entangled states that can be used as resources for quantum computing,quantum communication, quantum metrology, and other quantum informationprocessing tasks).

BACKGROUND

Quantum computers are computers that make use of quantum-mechanicalphenomena. In classical computing, information is represented as bits oflogical values. Quantum computing, in contrast, makes use of quantumbits (called “qubits”). While the state of a classical bit isconstrained to being one of the permitted logical values (e.g., a zeroor a one), qubits can make use of the quantum-mechanical phenomenon ofsuperposition, and thereby have a state that is a mixture of logicalvalues. Certain algorithms, such as Shor's prime factorizationalgorithm, take advantage of superposition and interference to speed upcomputational tasks. Thus, quantum computing promises a new paradigm ofcomputation where information is processed in a way that has noclassical analogue.

One of the main barriers to widespread use of quantum computing is thereliable generation and maintenance of resources. For example, manyquantum computing algorithms require clusters of qubits. These qubitsneed to be in a particular type of superposition, known as an entangledstate. However, various problems that either inhibit the generation ofentangled states or destroy the entanglement once created (e.g., such asde-coherence) have frustrated advancements in quantum computing.Accordingly, there is a need for methods and devices that generate andmaintain entangled states of qubits.

SUMMARY

Some embodiments described herein can use photons as the basis forqubits (e.g., each qubit is encoded in a degree of freedom of asingle-photon, such as the single-photon's polarization, or the degreesof freedom of a resource constructed from several single-photons) butother qubit types can be used without departing from the scope of thepresent disclosure. Using integrated optics, single-photons can be madeto have low de-coherence rates, thus solving the problem of maintenanceof entangled states once created. However, manipulation ofsingle-photons, such as generation of entangled states based onsingle-photons, is, in general, a probabilistic process rather than adeterministic one. For example, the probability of generating sixphotons in a Greenberger-Horn-Zeilinger (GHZ) state has been at most6.25%. Increasing the probability of generating a cluster of entangledphotons is greatly desired.

The above deficiencies and other related problems are reduced oreliminated by the methods and devices described herein for generatingentangled qubit states, e.g., for generating entangled photonic qubitsstates comprised of 2 or more photons. In particular, the embodimentsdescribed herein include methods and devices for generating entangledstates of photons that have a higher probability of success and usefewer photons as compared with conventional methods and devices forgenerating entangled qubit states. For example, the methods and devicesdescribed herein allow generating six photons in a GHZ state with aprobability of 18.75% in some configurations, which is three times themaximum probability previously known. Because generating single-photonsis itself a probabilistic process, using fewer photons to generate thesame size photonic state leads—by itself—to increased efficiency.

To that end, the present disclosure provides a method of obtaining aplurality of photons in a Greenberger-Horn-Zeilinger (GHZ) state. Themethod includes receiving a plurality of photon pairs, each photon pairbeing in a Bell state (e.g., an Einstein-Podolsky-Rosen pair) andincluding a first photon and a second photon that is distinct andseparate from the first photon. The method further includes obtaining aplurality of photons comprising at least six photons in a GHZ state byproviding photons of the plurality of photon pairs to a plurality ofbeam splitters. Providing the photons of the plurality of photon pairsto a plurality of beam splitters includes providing a first photon of afirst photon pair as a first input to a first beam splitter and a firstphoton of a second photon pair as a second input to the first beamsplitter. The first beam splitter is coupled with a first output channelof the first beam splitter and a second output channel of the first beamsplitter. Providing the photons of the plurality of photon pairs to aplurality of beam splitters further includes providing a first photon ofa third photon pair as a first input to a second beam splitter and afirst photon of a fourth photon pair as a second input to the secondbeam splitter that is distinct from the first beam splitter. The secondbeam splitter is coupled with a first output channel of the second beamsplitter and a second output channel of the second beam splitter. Themethod further includes providing a photon output from the first beamsplitter in the second output channel of the first beam splitter as afirst input to a first fusion gate and a photon output from the secondbeam splitter in the first output channel of the second beam splitter asa second input to the first fusion gate.

Further, the present disclosure provides a device for obtaining aplurality of photons in a Greenberger-Horn-Zeilinger (GHZ) state. Thedevice includes a first beam splitter coupled with: a first inputchannel for the first beam splitter configured for receiving a firstphoton of a photon pair from a first photon source of a plurality ofphoton sources; a second input channel for the first beam splitterconfigured for receiving a first photon of a photon pair from a secondphoton source of the plurality of photon sources; a first output channelfor the first beam splitter; and a second output channel for the firstbeam splitter. The device further includes a second beam splittercoupled with: a first input channel for the second beam splitterconfigured for receiving a first photon of a photon pair from a thirdphoton source of the plurality of photon sources; a second input channelfor the second beam splitter configured for receiving a first photon ofa photon pair from a fourth photon source of the plurality of photonsources; a first output channel for the second beam splitter; and asecond output channel for the second beam splitter. The device furtherincludes a first fusion gate coupled with: a first input channel, forthe first fusion gate, coupled with the second output channel of thefirst beam splitter; and a second input channel, for the first fusiongate, coupled with the first output channel of the first beam splitter.The device further includes respective input channels for receivingsecond photons from: the photon pair from the first photon source, thephoton pair from the second photon source, the photon pair from thethird photon source, and the photon pair from the fourth photon source.Each respective input channel for receiving a second photon is coupledwith a respective output channel.

BRIEF DESCRIPTION OF THE DRAWINGS

For a better understanding of the various described embodiments,reference should be made to the Detailed Description below, inconjunction with the following drawings in which like reference numeralsrefer to corresponding parts throughout the figures.

FIG. 1 shows a hybrid computing system in accordance with one or moreembodiments.

FIG. 2 shows a hybrid quantum computing system in accordance with someembodiments.

FIGS. 3A-3H are schematic diagrams illustrating devices for generatingentangled qubit states, in accordance with some embodiments.

FIGS. 4A-4B are schematic diagrams illustrating fusion gates, inaccordance with some embodiments.

FIG. 5 is a schematic diagram illustrating an interference-basedmicro-photonic beam splitter, in accordance with some embodiments.

FIG. 6 is a schematic diagram illustrating a device for generating Bellpairs, in accordance with some embodiments.

FIGS. 7A-7C illustrate a flow chart for a method of generating entangledqubit states, in accordance with some embodiments.

FIG. 8 is a schematic diagram illustrating an architecture of a devicefor generating entangled qubit states, in accordance with someembodiments.

FIG. 9 shows schematic diagrams of beamsplitters and Hadamard gates inaccordance with some embodiments.

FIGS. 10A-10C illustrate schematic diagrams of waveguide beam splitters,in accordance with some embodiments.

FIG. 11 illustrates a mapping between photonic components for use in thepolarization encoding and for use in the path encoding, in accordancewith some embodiments.

FIG. 12 is a schematic diagram illustrating devices for generatingentangled qubit states, in accordance with some embodiments.

DETAILED DESCRIPTION

Reference will now be made in detail to implementations, examples ofwhich are illustrated in the accompanying drawings. In the followingdetailed description, numerous specific details are set forth in orderto provide a thorough understanding of the various describedimplementations. However, it will be apparent to one of ordinary skillin the art that the various described implementations may be practicedwithout these specific details. In other instances, well-known methods,procedures, components, circuits, and networks have not been describedin detail so as not to unnecessarily obscure aspects of theimplementations.

Many modifications and variations of this disclosure can be made withoutdeparting from its spirit and scope, as will be apparent to thoseskilled in the art. The specific implementations described herein areoffered by way of example only, and the disclosure is to be limited onlyby the terms of the appended claims, along with the full scope ofequivalents to which such claims are entitled.

I. Introduction to Qubits and Path Encoding

The dynamics of quantum objects, e.g., photons, electrons, atoms, ions,molecules, nanostructures, and the like, follow the rules of quantumtheory. More specifically, in quantum theory, the quantum state of aquantum object, e.g., a photon, is described by a set of physicalproperties, the complete set of which is referred to as a mode. In someembodiments, a mode is defined by specifying the value (or distributionof values) of one or more properties of the quantum object. For example,again for photons, modes can be defined by the frequency of the photon,the position in space of the photon (e.g., which waveguide orsuperposition of waveguides the photon is propagating within), theassociated direction of propagation (e.g., the k-vector for a photon infree space), the polarization state of the photon (e.g., the direction(horizontal or vertical) of the photon's electric and/or magneticfields) and the like.

For the case of photons propagating in a waveguide, it is convenient toexpress the state of the photon as one of a set of discretespatio-temporal modes. For example, the spatial mode k_(i) of the photonis determined according to which one of a finite set of discretewaveguides the photon can be propagating in. Furthermore, the temporalmode t_(j) is determined by which one of a set of discrete time periods(referred to herein as “bins”) the photon can be present in. In someembodiments, the temporal discretization of the system can be providedby the timing of a pulsed laser which is responsible for generating thephotons. In the examples below, spatial modes will be used primarily toavoid complication of the description. However, one of ordinary skillwill appreciate that the systems and methods can apply to any type ofmode, e.g., temporal modes, polarization modes, and any other mode orset of modes that serves to specify the quantum state. Furthermore, inthe description that follows, embodiments will be described that employphotonic waveguides to define the spatial modes of the photon. However,one of ordinary skill having the benefit of this disclosure willappreciate that any type of mode, e.g., polarization modes, temporalmodes, and the like, can be used without departing from the scope of thepresent disclosure.

For quantum systems of multiple indistinguishable particles, rather thandescribing the quantum state of each particle in the system, it isuseful to describe the quantum state of the entire many-body systemusing the formalism of Fock states (sometimes referred to as theoccupation number representation). In the Fock state description, themany-body quantum state is specified by how many particles there are ineach mode of the system. Because modes are the complete set ofproperties, this description is sufficient. For example, a multi-mode,two particle Fock state |1001

_(1,2,3,4) specifies a two-particle quantum state with one photon inmode 1, zero photons in mode 2, zero photons in mode three, and 1 photonin mode four. Again, as introduced above, a mode can be any set ofproperties of the quantum object (and can depend on the single particlebasis states being used to define the quantum state). For the case ofthe photon, any two modes of the electromagnetic field can be used,e.g., one may design the system to use modes that are related to adegree of freedom that can be manipulated passively with linear optics.For example, polarization, spatial degree of freedom, or angularmomentum, could be used. For example, the four-mode system representedby the two particle Fock state |1001

_(1,2,3,4) can be physically implemented as four distinct waveguideswith two of the four waveguides (representing mode 1 and mode 4,respectively) having one photon travelling within them. Other examplesof a state of such a many-body quantum system are the four photon Fockstate |1111

_(1,2,3,4) that represents each waveguide containing one photon and thefour photon Fock state |2200

_(1,2,3,4) that represents waveguides one and two respectively housingtwo photons and waveguides three and four housing zero photons. Formodes having zero photons present, the term “vacuum mode” is used. Forexample, for the four photon Fock state |2200

_(1,2,3,4) modes 3 and 4 are referred to herein as “vacuum modes” (alsoreferred to as “ancilla modes”).

As used herein, a “qubit” (or quantum bit) is a physical quantum systemwith an associated quantum state that can be used to encode information.Qubits, in contrast to classical bits, can have a state that is asuperposition of logical values such as 0 and 1. In some embodiments, aqubit is “dual-rail encoded” such that the logical value of the qubit isencoded by occupation of one of two modes by exactly one photon (asingle photon). For example, consider the two spatial modes of aphotonic system associated with two distinct waveguides. In someembodiments, the logical 0 and 1 values can be encoded as follows:|0

_(L)=|10

_(1,2)  (1)|1

_(L)=|01

_(1,2)  (2)where the subscript “L” indicates that the ket represents a logicalvalue (e.g., a qubit value) and, as before, the notation |ij

_(1,2) on the right-hand side of the Equations (1)-(2) above indicatesthat there are i photons in a first waveguide and j photons in a secondwaveguide, respectively (e.g., where i and j are integers). In thisnotation, a two qubit state having a logical value |01

_(L) (representing a state of two qubits, the first qubit being in a ‘0’logical state and the second qubit being in a ‘1’ logical state) may berepresented using photon occupations across four distinct waveguides by|1001

_(1,2,3,4) (i.e., one photon in a first waveguide, zero photons in asecond waveguide, zero photons in a third waveguide, and one photon in afourth waveguide). In some instances, throughout this disclosure, thevarious subscripts are omitted to avoid unnecessary mathematicalclutter.

A Bell pair is a pair of qubits in any type of maximally entangled statereferred to as a Bell state. For dual rail encoded qubits, examples ofBell states include:

$\begin{matrix}{\left. \left| \Phi^{+} \right. \right\rangle = {\frac{\left. \left. {\left. \left. \left| 0 \right. \right\rangle_{L} \middle| 0 \right\rangle_{L} +} \middle| 1 \right\rangle_{L} \middle| 1 \right\rangle_{L}}{\sqrt{2}} = \frac{\left. {\left. \left| {1010} \right. \right\rangle +} \middle| {0101} \right\rangle}{\sqrt{2}}}} & (3)\end{matrix}$ $\begin{matrix}{\left. \left| \Phi^{-} \right. \right\rangle = {\frac{\left. \left. {\left. \left. \left| 0 \right. \right\rangle_{L} \middle| 0 \right\rangle_{L} -} \middle| 1 \right\rangle_{L} \middle| 1 \right\rangle_{L}}{\sqrt{2}} = \frac{\left. {\left. \left| {1010} \right. \right\rangle -} \middle| {0101} \right\rangle}{\sqrt{2}}}} & (4)\end{matrix}$ $\begin{matrix}{\left. {❘\Psi^{+}} \right\rangle = {\frac{\left. \left. {\left. \left. \left| 0 \right. \right\rangle_{L} \middle| 1 \right\rangle_{L} +} \middle| 1 \right\rangle_{L} \middle| 0 \right\rangle_{L}}{\sqrt{2}} = \frac{\left. {\left. \left| {1001} \right. \right\rangle +} \middle| {0110} \right\rangle}{\sqrt{2}}}} & (5)\end{matrix}$ $\begin{matrix}{\left. \left| \Psi^{-} \right. \right\rangle = {\frac{\left. \left. {\left. \left. \left| 0 \right. \right\rangle_{L} \middle| 1 \right\rangle_{L} -} \middle| 1 \right\rangle_{L} \middle| 0 \right\rangle_{L}}{\sqrt{2}} = \frac{\left. {\left. \left| {1001} \right. \right\rangle -} \middle| {0110} \right\rangle}{\sqrt{2}}}} & (6)\end{matrix}$

In a computational basis (e.g., logical basis) with two states, aGreenberger-Horne-Zeilinger state is a quantum superposition of allqubits being in a first state of the two states superposed with all ofqubits being in a second state. Using logical basis described above, thegeneral M-qubit GHZ state can be written as:

$\begin{matrix}{\left. {❘{GHZ}} \right\rangle = \frac{\left. {{{\left. {❘0} \right\rangle^{\otimes M} +}❘}1} \right\rangle^{\otimes M}}{\sqrt{2}}} & (7)\end{matrix}$

II. A Hybrid Classical-Quantum Computing System

FIG. 1 shows a hybrid computing system in accordance with one or moreembodiments. The hybrid computing system 101 includes a user interfacedevice 104 that is communicatively coupled to a hybrid quantum computing(QC) sub-system 106, described in more detail below in FIG. 2 . The userinterface device 104 can be any type of user interface device, e.g., aterminal including a display, keyboard, mouse, touchscreen and the like.In addition, the user interface device can itself be a computer such asa personal computer (PC), laptop, tablet computer and the like. In someembodiments, the user interface device 104 provides an interface withwhich a user can interact with the hybrid QC subsystem 106. For example,the user interface device 104 may run software, such as a text editor,an interactive development environment (IDE), command prompt, graphicaluser interface, and the like so that the user can program, or otherwiseinteract with, the QC subsystem to run one or more quantum algorithms.In other embodiments, the hybrid QC subsystem 106 may be pre-programmedand the user interface device 104 may simply be an interface where auser can initiate a quantum computation, monitor the progress, andreceive results from the hybrid QC subsystem 106. Hybrid QC subsystem106 further includes a classical computing system 108 coupled to one ormore quantum computing chips 110. In some examples, the classicalcomputing system 108 and the quantum computing chips 110 can be coupledto other electronic and/or optical components, e.g., pulsed pump lasers,microwave oscillators, power supplies, networking hardware, etc. In someembodiments that require cryogenic operation, the quantum computingchips 110 can be housed within a cryostat, e.g., cryostat 114. In otherembodiments for which cryogenic operation is not required, the quantumcomputing chips 110, the cryostat 114 may be replaced with any otherenclosure. In some embodiments, the quantum computing chips 110 caninclude one or more constituent chips, e.g., hybrid control electronics116 and integrated photonics chip 118. Signals can be routed on- andoff-chip any number of ways, e.g., via optical interconnects 120 and viaother electronic interconnects 122. In addition, the hybrid computingsystem 101 may employ a quantum computing process, e.g.,measurement-based quantum computing (MBQC), circuit-based quantumcomputing (CBQC) or any other quantum computing scheme.

FIG. 2 shows a block diagram of a hybrid QC system 201 in accordancewith some embodiments. Such a system can be associated with the hybridcomputing system 101 introduced above in reference to FIG. 1 . In FIG. 2, solid lines represent quantum information channels and dashedrepresent classical information channels. The hybrid QC system 201includes a qubit entangling system 203, qubit readout circuit 205, andclassical computing system 207. In some embodiments, the qubitentangling system 203 takes as input a collection of N physical qubits,e.g., physical qubits 209 (also represented schematically as inputs 211a, 211 b, 211 c, . . . , 211 n) and generates quantum entanglementbetween two or more of them to generate an entangled state 215. Forexample, in the case of photonic qubits, the qubit entangling system 203can be a linear optical system such as an integrated photonic circuitthat includes waveguides, beam splitters, photon detectors, delay lines,and the like. In some examples, the entangled state 215 can be alattice, cluster, or graph state, or one part of a larger lattice,cluster, or graph state that is created over the course of several clockcycles of the quantum computer. In some embodiments, the physical qubits209 can be a collection of quantum systems and/or particles and can beformed using any qubit architecture. For example, the quantum systemscan be particles such as atoms, ions, nuclei, and/or photons. In otherexamples, the quantum systems can be other engineered quantum systemssuch as flux qubits, phase qubits, or charge qubits (e.g., formed from asuperconducting Josephson junction), topological qubits (e.g., Majoranafermions), or spin qubits formed from vacancy centers (e.g., nitrogenvacancies in diamond). Furthermore, for the sake of clarity ofdescription, the term “qubit” is used herein although the system canalso employ quantum information carriers that encode information in amanner that is not necessarily associated with a binary bit. Forexample, qudits can be used, i.e., quantum systems that can encodeinformation in more than two quantum states in accordance with someembodiments.

In accordance with some embodiments, the hybrid QC system 201 can be aquantum circuit-based quantum computer or a measurement-based quantumcomputer. In either case, a software program (e.g., a set of machinereadable instructions) that represents the quantum algorithm to be runon the hybrid QC system 201 can be passed to a classical computingsystem 207 (e.g., corresponding to classical computing system 107 inFIG. 1 above). The classical computing system 207 can be any type ofcomputing device such as a PC, one or more blade servers, and the like,or even a high-performance computing system such as a supercomputer,server farm, and the like. Such a system can include one or moreprocessors (not shown) coupled to one or more computer memories, e.g.,memory 206. Such a computing system will be referred to herein as a“classical computer.” In some examples, the software program can bereceived by a classical computing module, referred to herein as adetection pattern generator 213. One function of the detection patterngenerator 213 is to generate a set of machine-level instructions fromthe input software program (which may originate as code that can be moreeasily written by a user to program the quantum computer), i.e., thedetection pattern generator 213 operates as a compiler for softwareprograms to be run on the quantum computer. Detection pattern generator213 can be implemented as pure hardware, pure software, or anycombination of one or more hardware or software components or modules.In some examples, the compiled machine-level instructions take the formof one or more data frames that instruct the qubit readout circuit tomake one or more quantum measurements on the entangled state 215.Measurement pattern 217 (e.g., a data frame) is one example of the setof measurements that should be applied to the individual qubits ofentangled state 215 during a certain clock cycle as the program isexecuted. In some embodiments, several measurement patterns 217 can bestored in memory 206 as classical data. Generally, the measurementpatterns 217 can dictate whether or not a detector from the qubitdetection array 221 of the qubit readout circuit 205 should make ameasurement on a given qubit that makes up the entangled state 215. Inaddition, the measurement pattern 217 can also store which basis (e.g.,Pauli X, Y, Z, etc.) the measurement should be made in order to executethe program. In some examples, the measurement pattern 217 can alsoinclude a set of gates that should be applied by the qubit entanglingcircuit to the next set of physical qubits 209 that are to be processedat some future clock cycle of the hybrid QC system 201.

A controller 219 of the qubit readout circuit 205 can receive data thatencodes the measurement pattern 217 and generate the configurationsignals necessary to drive a set of detectors within the qubit detectionarray 221. The detectors can be any detector that can detect the quantumstates of one or more of the qubits in the entangled state 215. Forexample, for the case of photonic qubits, the detectors can be singlephoton detectors that are coupled to one or more waveguides, beamsplitters, interferometers, switches, polarizers, polarization rotatorsand the like. One of ordinary skill will appreciate that many types ofdetectors may be used depending on the particular qubit architecture.

In some embodiments, the result of applying the measurement pattern 217(e.g. a detection pattern) to the qubit detection array is a readoutoperation that “reads out” the quantum states of the qubits in theentangled state 215. Once this measurement is accomplished, the quantuminformation stored within the entangled state 215 is converted toclassical information that corresponds to a set of eigenvalues that aremeasured by the detectors, referred to herein as “measurement outcomes.”These measurement outcomes can be stored in a measurement outcome dataframe, e.g., data frame 222 and passed back to the classical computingsystem for further processing.

In some embodiments, any of the submodules in the hybrid QC system 201,e.g., controller 223, quantum gate array 225, qubit detection array 221,controller 219, detection pattern generator 213, decoder 233, andlogical processor 208 can include any number of classical computingcomponents such as processors (CPUs, GPUs, TPUs) memory (any form ofRAM, ROM), hard coded logic components (classical logic gates such asAND, OR, XOR, etc.) and/or programmable logic components such as fieldprogrammable gate arrays (FPGAs and the like). These modules can alsoinclude any number of application specific integrated circuits (ASICs),microcontrollers (MCUs), systems on a chip (SOCs), and other similarmicroelectronics.

In some embodiments, the entangled state 215 can be any cluster statedescribed herein. As described herein, the logical qubit measurementoutcomes 227 can be fault tolerantly recovered, e.g., via decoder 233,from the data frames 222 of the physical qubits. Logical processor 208can then process the logical outcomes as part of the running of theprogram. As shown, the logical processor can feed back information tothe detection pattern generator 213 to affect downstream gates and/ormeasurements to ensure that the computation proceeds fault tolerantly.

In the description that follows, embodiments will be described thatemploy spatial modes of photons as the qubit system, but one of ordinaryskill will appreciate that any type of qubit described by any type ofmode can be employed without departing from the scope of the presentdisclosure. Furthermore, in what follows, photonic waveguides are usedto define the spatial modes of the photon. However, one of ordinaryskill having the benefit of this disclosure will appreciate that anytype of mode, e.g., polarization modes, temporal modes, and the like,can be used without departing from the scope of the present disclosure.The diagrams shown in the remaining figures are schematic diagrams witheach horizontal line representing a mode of a quantum system, e.g., awaveguide.

As used herein, the term “quantum computing resource” is intended toinclude algorithmic resources as well as precursor resources (e.g.,resources that are combined with other resources to form a largerresource, which may be an algorithmic resource). Algorithmic resourcesmay include universal quantum computing resource states and/orfault-tolerant computational resource states.

III. Generation of Entangled Qubit States

FIGS. 3A-3H are schematic diagrams illustrating devices 300 forgenerating entangled qubit states, in accordance with some embodiments.

FIGS. 3A-3B are schematic diagrams of a device 300-a for generatingentangled qubit states in accordance with some embodiments. FIG. 3A andFIG. 3B illustrate the same device, but show different reference numbersfor visual clarity. Thus, FIGS. 3A-3B should be viewed together.

Device 300-a probabilistically generates (or provides) six qubits in aGHZ state (also called herein a 6-photon GHZ state) from 4 pairs ofqubits (i.e., 8 qubits), each pair in a Bell state (also called herein 4Bell pairs). Examples of n-GHZ states and Bell states are shown above(for the dual rail encoding) in Eqns. (1)-(7). The dual rail encodinghas a one-to-one correspondence with the polarization encoding per thefollowing definitions:|0

_(L)=|10

_(1,2) →|h>  (8)|1

_(L)=|01

_(1,2) →|v>,  (9)with the choice of which logical state (|0> or |1>) maps to whichpolarization/dual rail encoded state being an arbitrary choice, i.e., hand v in equations (8) and (9) above can be swapped without departingfrom the scope of the present disclosure.

When successful, the 6-qubit GHZ state includes a qubit at each outputchannel 320 (e.g., a single-qubit at each output channel 320). The qubitat each output channel 320 is entangled with the qubit at each otheroutput channel 320 of device 300-a such that the output qubits are inthe GHZ state (e.g., the set of output qubits is in amaximally-entangled state). More specifically the 6-qubit GHZ stateincludes qubits 308-a 2, 308-b 2, 308-c 2, 320-d 2, and the tworemaining output qubits that our output at each output channel 320.

In some embodiments, as described below, device 300-a can be implementedwith photonic qubits, i.e., qubits that are photons in which case thedevice 300-a includes two distinct and separate beam splitters and asingle fusion gate. Fusion gates are described in more detail below inreference to FIGS. 4A and 4B. A path encoded version of device 300-a ina photonic qubit implementation is shown in FIG. 12 . At least onephoton output from each beam splitter of the two distinct and separatebeam splitters is provided to the fusion gate (e.g., each of the twodistinct and separate beam splitters provides a photon to the fusiongate). In some embodiments, each of the photons received by the fusiongate is provided by an output from a beam splitter (e.g., a distinctbeam splitter). In some embodiments, the fusion gate receives photonsfrom each of the two distinct beam splitters.

Device 300-a includes a first beam splitter 302-a (e.g., a photonicinterference-based beam splitter, FIG. 5 ) coupled with a first inputchannel 304-a for the first beam splitter 302-a. The first input channel304-a for the first beam splitter 302-a is configured for receiving afirst photon 308-a 1 of a photon pair 308-a (e.g., a pair of photons308-a 1 and 308-a 2) from a first photon source (not shown) (e.g., arespective Bell pair source, FIG. 6 ) of a plurality of photon sources(not shown).

In some embodiments, the beam splitters 302 include a polarizationrotator at each input and output. For visual clarity, these polarizationrotators are not shown in FIGS. 3A-3H, but their operation is describedwith reference to method 700.

The first beam splitter 302-a is further coupled with a second inputchannel 306-a for the first beam splitter 302-a. The second inputchannel 306-a for the first beam splitter 302-a is configured forreceiving a first photon 308-b 1 of a photon pair 308-b (e.g., a pair ofphotons 308-b 1 and 308-b 2) from a second photon source (e.g., arespective Bell pair source, FIG. 6 ) of the plurality of photonsources.

The first beam splitter 302-a is further coupled with a first outputchannel 310-a for the first beam splitter 302-a and a second outputchannel 312-a for the first beam splitter 302-a.

In some embodiments, a first channel and a second channel are respectiveportions of a single (typically larger and/or longer) channel. Forexample, first input channels 304 and first output channels 310 may beportions (e.g., lengths or segments) of a single micro-photonic (e.g.,integrated) channel fabricated on a chip. Similarly, second inputchannels 306 and second output channels 312 may be portions (e.g.,lengths or segments) of a single micro-photonic (e.g., integrated)channel fabricated on a chip.

Device 300-a further includes a second beam splitter 302-b (e.g., aphotonic interference-based beam splitter, FIG. 5 ) coupled with a firstinput channel 304-b for the second beam splitter 302-b. The first inputchannel 304-b for the second beam splitter 302-b is configured forreceiving a first photon 308-c 1 of a photon pair 308-c (e.g., a pair ofphotons 308-c 1 and 308-c 2) from a third photon source (e.g., arespective Bell pair source, FIG. 6 ) of the plurality of photonsources.

The second beam splitter 302-b is further coupled with a second inputchannel 306-b for the second beam splitter 302-b. The second inputchannel 306-b for the second beam splitter 302-b is configured forreceiving a first photon 308-d 1 of a photon pair 308-d (e.g., a pair ofphotons 308-d 1 and 308-d 2) from a fourth photon source (e.g., arespective Bell pair source, FIG. 6 ) of the plurality of photonsources.

The second beam splitter 302-b is further coupled with a first outputchannel 310-b for the second beam splitter 302-b and a second outputchannel 312-b for the second beam splitter 302-b.

Device 300-a further includes a first fusion gate 314-a. In someembodiments, the first fusion gate 314-a is a Type-II fusion gate, asdescribed below with reference to FIG. 4B.

The first fusion gate 314-a is coupled with a first input channel 316-afor the first fusion gate 314-a. The first input channel 316-a for thefirst fusion gate 314-a is coupled with the second output channel 312-aof the first beam splitter 302-a. The first fusion gate 314-a is furthercoupled with a second input channel 318-a for the first fusion gate314-a. The second input channel 318-a for the first fusion gate 314-a iscoupled with the first output channel 310-b of the second beam splitter302-b.

In some embodiments, as shown in FIGS. 3A and 3B, device 300-a furtherincludes: input channel 322-a for receiving a second photon 308-a 2 ofthe photon pair 308-a from the first photon source (e.g., a Bell pairsource); input channel 322-b for receiving a second photon 308-b 2 ofthe photon pair 308-b from the second photon source (e.g., a Bell pairsource); input channel 322-c for receiving a second photon 308-c 2 ofthe photon pair 308-c from the third photon source (e.g., a Bell pairsource); and input channel 322-d for receiving a second photon 308-d 2of the photon pair 308-d from the fourth photon source (e.g., a Bellpair source).

Each respective input channel 322 for receiving a second photon iscoupled with a respective output channel 320. For example, input channel322-a is coupled with output channel 320-a; input channel 322-b iscoupled with output channel 320-b; input channel 322-c is coupled withoutput channel 320-c; and input channel 322-d is coupled with outputchannel 320-d. The photons provided to output channels 320 (e.g., secondphotons 308-a 2, 308-b 2, 308-c 2, 308-d 2, and the photons provided tooutput channels 320-e and 320-f) are, at least probabilistically, in a6-photon GHZ state after the first fusion gate 314-a operates on thephotons it receives. In some embodiments, the outcome of the operationof the first fusion gate 314-a (e.g., detection of two photons) is usedto determine whether the photons provided to output channels 320 are ina 6-photon GHZ state.

Again, respective output channels 320 and the corresponding channelscoupled with the respective output channels 320 (e.g., input channels322 and/or output channels 316/318) may be portions (e.g., lengths orsegments) of a single channel (e.g., a micro-photonic and/or integratedchannel fabricated on a chip). Likewise, output channels 310/312 and thecorresponding input channels 316/318 coupled with output channels310/312 may be portions (e.g., lengths or segments) of a single channel(e.g., a micro-photonic and/or integrated channel fabricated on a chip).

FIGS. 3C-3D are schematic diagrams of a device 300-b for generatingentangled qubit states in accordance with some embodiments. FIG. 3C andFIG. 3D illustrate the same device, but show different reference numbersfor visual clarity. Thus, FIGS. 3C-3D should be viewed together.

Device 300-b probabilistically generates (or provides) eight photons ina GHZ state (also called herein an 8-photon GHZ state) from 6 pair ofphotons (i.e., 12 photons), each pair in a Bell state (also calledherein 6 Bell pairs). When successful, the 8-photon GHZ state includes aphoton at each output channel 320 (e.g., a single-photon at each outputchannel 320). The photon at each output channel 320 is entangled withthe photon at each other output channel 320 of device 300-b such thatthe output photons are in a GHZ state (e.g., the set of output photonsis maximally entangled).

Device 300-b is analogous to device 300-a, but with the followingdifferences and additions.

In some embodiments, as described below, device 300-b includes aplurality of (e.g., three) distinct and separate beam splitters and aplurality of (e.g., two) fusion gates (e.g., Type II fusion gates). Atleast one photon output from each beam splitter of the three distinctand separate beam splitters is provided to one of the plurality offusion gates (e.g., each of the three distinct and separate beamsplitters provides a photon to a respective fusion gate of the pluralityof fusion gates). In some embodiments, each of the photons received bythe plurality of fusion gates is provided by an output of a beamsplitter (e.g., none of the photons received by the respective fusiongate of the plurality of fusion gates is received directly from a Bellpair generator). In some embodiments, each fusion gate of the pluralityof fusion gates receives photons from two distinct beam splitters of theplurality of beam splitters.

Device 300-b includes a third beam splitter 302-c (e.g., a photonicinterference-based beam splitter, FIG. 5 ). The third beam splitter302-c is coupled with a first input channel 304-c for the third beamsplitter 302-c. The first input channel 304-c is configured forreceiving a first photon 308-e 1 of a photon pair 308-e (e.g., a pair ofphotons 308-e 1 and 308-e 2) from a fifth photon source (e.g., arespective Bell pair source, FIG. 6 ) of the plurality of photonsources. The third beam splitter 302-c is further coupled with a secondinput channel 306-c for the third beam splitter 302-c. The second inputchannel 306-c is configured for receiving a first photon 308-f 1 of aphoton pair 308-f (e.g., a pair of photons 308-f 1 and 308-f 2) from asixth photon source (e.g., a respective Bell pair source, FIG. 6 ) ofthe plurality of photon sources.

The third beam splitter 302-c is further coupled with a first outputchannel 310-c for the third beam splitter 302-c and a second outputchannel 312-c for the third beam splitter 302-c. Device 300-b furtherincludes a second fusion gate 314-b. In some embodiments, the secondfusion gate 314-b is a Type-II fusion gate, as described below withreference to FIG. 4B.

The second fusion gate 314-b is coupled with a first input channel 316-bfor the second fusion gate 314-b. The first input channel 316-b for thesecond fusion gate 314-b is coupled with the second output channel 312-bof the second beam splitter 302-b. The second fusion gate 314-b isfurther coupled with a second input channel 318-b for the second fusiongate 314-b. The second input channel 318-b for the second fusion gate314-b is coupled with the first output channel 310-c of the third beamsplitter 302-c.

Device 300-b further includes: input channel 322-e for receiving asecond photon 308-e 2 from the photon pair 308-e from the fifth photonsource and input channel 322-f for receiving a second photon 308-f 2from the photon pair 308-f from the sixth photon source.

Each respective input channel 322 for receiving a second photon iscoupled with a respective output channel 320. For example, in additionto those input channels described with reference to device 300-a, inputchannel 322-e is coupled with output channel 320-g and input channel322-f is coupled with output channel 320-h. The photons provided tooutput channels 320 (e.g., second photons 308-a 2, 308-b 2, 308-c 2,308-d 2, 308-e 2, 308-f 2 and the photons received via channels 310-aand 312-c) are, at least probabilistically, in an 8-photon GHZ stateafter the first fusion gate 314-a operates on the photons it receivesand the second fusion gate 314-b operates on the photons it receives. Insome embodiments, the outcome of the operation of the first fusion gate314-a (e.g., detection of two photons) and the operation of the secondfusion gate 314-b (e.g., detection of two photons) are used to determinewhether the photons provided to output channels 320 are in an 8-photonGHZ state.

FIG. 3E is a schematic diagram of a device 300-c for generatingentangled qubit states, in accordance with some embodiments. Device300-c is analogous to device 300-b (FIGS. 3C-3D) except that FIG. 3Eincludes a repeating unit 324 which is repeated n times, where n isgreater than or equal to 1 (e.g., in some configurations, device 300-cincludes three fusion gates 314). Device 300-c probabilisticallygenerates a (2n+6)-photon GHZ state (e.g., (2n+6) photons in a GHZstate) from (2n+4) pairs of photons (i.e., 4n+8 photons), each pair in aBell state (also called herein (2n+4) Bell pairs). When successful, the(2n+6)-photon GHZ state includes a photon at each output channel 320(e.g., a single-photon at each output channel 320). The photon at eachoutput channel 320 is entangled with the photon at each other outputchannel 320 of device 300-c such that the output photons are in a GHZstate (e.g., the set of output photons is in a maximally-entangledstate).

Device 300-b is an example of device 300-c with n=1. Thus, FIG. 3Eillustrates that device 300-b can be extended to form an arbitrarilylarge GHZ state (e.g., a GHZ state having an arbitrarily large evennumber of photons), albeit with decreasing probability of success as thesize of the GHZ state increases (e.g., because the outcomes measured byeach of the fusion gates must be indicative of success, which becomesless likely as there are more fusion gates corresponding to morenecessary and independent indicia of success).

Another example is n=2, in which case device 300-c generates a 10-photonGHZ state. In that example, device 300-c includes a fourth beamsplitter. The fourth beam splitter is coupled with a first input channelfor the fourth beam splitter. The first input channel for the fourthbeam splitter is configured for receiving a first photon of a photonpair from a seventh photon source (e.g., a respective Bell pair source,FIG. 6 ) of the plurality of photon sources. The fourth beam splitter iscoupled with a second input channel for the fourth beam splitter. Thesecond input channel for the fourth beam splitter is configured forreceiving a first photon of a photon pair from an eighth photon source(e.g., a respective Bell pair source, FIG. 6 ) of the plurality ofphoton sources. The n=2 example of device 300-c includes a first outputchannel for the fourth beam splitter and a second output channel for thefourth beam splitter. The n=2 example of device 300-c further includes athird fusion gate (e.g., a third Type-II fusion gate) coupled with afirst input channel for the third fusion gate. The first input channelfor the third fusion gate is coupled with the second output channel ofthe third beam splitter. The third fusion gate is coupled with a secondinput channel for the third fusion gate. The second input channel forthe third fusion gate is coupled with the first output channel of thefourth beam splitter. Device 300-c respective input channels forreceiving second photons from: the photon pair from the seventh photonsource and the photon pair from the eighth photon source. Eachrespective input channel for receiving a second photon is coupled with arespective output channel.

As shown below, the n=2 example of device 300-c is similar to device300-d illustrated in FIGS. 3F and 3G, but with three Type-II fusiongates rather than a Type-I fusion gate sandwiched between two Type-IIfusion gates.

FIGS. 3F-3G are schematic diagrams of a device 300-d for generatingentangled qubit states in accordance with some embodiments. FIG. 3F andFIG. 3G illustrate the same device, but show different reference numbersfor visual clarity. Thus, FIGS. 3F-3G should be viewed together.

Device 300-d probabilistically generates an 11-photon GHZ state from 8Bell pairs (i.e., 16 photons). When successful, the 11-photon GHZ stateincludes a photon at each output channel 320. The photon at each outputchannel 320 is entangled with the photon at each other output channel320 of the device 300-d such that the output photons are in a GHZ state(e.g., the set of output photons is in a maximally-entangled state).

Device 300-d is analogous to device 300-a, but with the followingdifferences and additions.

Device 300-d includes a third beam splitter 302-c (e.g., a photonic beamsplitter, FIG. 5 ). The third beam splitter 302-c is coupled with afirst input channel 304-c for the third beam splitter 302-c. The firstinput channel 304-c is configured for receiving a first photon 308-e 1of a photon pair 308-e from a fifth photon source (e.g., a respectiveBell pair source, FIG. 6 ) of the plurality of photon sources. The thirdbeam splitter 302-c is further coupled with a second input channel 306-cfor the third beam splitter 302-c. The second input channel 306-c isconfigured for receiving a first photon 308-f 1 of a photon pair 308-ffrom a sixth photon source (e.g., a respective Bell pair source, FIG. 6) of the plurality of photon sources.

The third beam splitter 302-c is further coupled with a first outputchannel 310-c for the third beam splitter 302-c and a second outputchannel 312-c for the third beam splitter 302-c.

Device 300-d further includes a second fusion gate 314-c. In someembodiments, the second fusion gate 314-c is a Type-I fusion gate, asdescribed below with reference to FIG. 4A.

The second fusion gate 314-c is coupled with a first input channel 316-cfor the second fusion gate 314-c. The first input channel 316-c for thesecond fusion gate 314-c is coupled with the second output channel 312-bof the second beam splitter 302-b. The second fusion gate 314-c isfurther coupled with a second input channel 318-c for the second fusiongate 314-c. The second input channel 318-c for the second fusion gate314-c is coupled with the first output channel 310-c of the third beamsplitter 302-c.

Device 300-d further includes a fourth beam splitter 302-d (e.g., aphotonic interference-based beam splitter, FIG. 5 ). The fourth beamsplitter 302-d is coupled with a first input channel 304-d for thefourth beam splitter 302-d. The first input channel 304-d is configuredfor receiving a first photon 308-g 1 of a photon pair 308-g from aseventh photon source (e.g., a respective Bell pair source, FIG. 6 ) ofthe plurality of photon sources.

The fourth beam splitter 302-d is coupled with a second input channel306-d for the fourth beam splitter 302-d. The second input channel 306-dis configured for receiving a first photon 308-h 1 of a photon pair308-h from an eighth photon source (e.g., a respective Bell pair source,FIG. 6 ) of the plurality of photon sources.

The fourth beam splitter 302-d is coupled with a first output channel310-d for the fourth beam splitter 302-d and second output channel 312-dfor the fourth beam splitter 302-d.

Device 300-d further includes a third fusion gate 314-d. The thirdfusion gate 314-d is coupled with a first input channel 316-d for thethird fusion gate 314-d. The first input channel 316-d for the thirdfusion gate 314-d is coupled with the second output channel 312-c of thethird beam splitter. The third fusion gate 314-d is coupled with asecond input channel 318-d for the third fusion gate 314-d. The secondinput channel 318-d for the third fusion gate 314-d is coupled with thefirst output channel 310-d of the fourth beam splitter 302-d.

Device 300-d further includes: input channel 322-e for receiving asecond photon 308-e 2 from the photon pair 308-e from the fifth photonsource; input channel 322-f for receiving a second photon 308-f 2 fromthe photon pair 308-f from the sixth photon source; input channel 322-gfor receiving a second photon 308-g 2 from the photon pair 308-g fromthe seventh photon source; and input channel 322-h for receiving asecond photon 308-h 2 from the photon pair 308-h from the eighth photonsource.

Each respective input channel 322 for receiving a second photon iscoupled with a respective output channel 320. For example, in additionto those input channels described with reference to device 300-a: inputchannel 322-e is coupled with output channel 320-g; input channel 322-fis coupled with output channel 320-h; input channel 322-g is coupledwith output channel 320-i; and input channel 322-h is coupled withoutput channel 320-j.

The photons provided to output channels 320 (e.g., second photons 308-a2, 308-b 2, 308-c 2, 308-d 2, 308-e 2, 308-f 2, 308-g 2, 308-h 2 and thephotons received by output channels 320-e, 320-k, and 320-f) are, atleast probabilistically, in an 11-photon GHZ state (e.g., 11 photons ina GHZ state) after the first fusion gate 314-a operates on the photonsit receives, the second fusion gate 314-c operates on the photons itreceives; and the third fusion gate 314-d operates on the photons itreceives. In some embodiments, the outcome of the operation of the firstfusion gate 314-a (e.g., detection of two photons), the outcome of theoperation of the second fusion gate 314-c (e.g., detection of onephoton), and the outcome of the operation of the third fusion gate 314-d(e.g., detection of two photons) are used to determine whether thephotons provided to output channels 320 are in an 11-photon GHZ state.

FIG. 3H is a schematic diagram of a device 300-e for generatingentangled qubit states in accordance with some embodiments. Device 300-eis analogous to device 300-d (FIGS. 3F-3G) except that FIG. 3H includesa repeating unit 326 which is repeated n times, where n is greater thanor equal to 1. The repeating unit comprises a Type-I fusion gate and aType-II fusion gate (along with their inputs and outputs, as previouslydescribed), such that each Type-I fusion gate is sandwiched between twoType-II fusion gates in the repeating structure.

Device 300-e probabilistically generates a (4(n+1)+n+2)—photon GHZ statefrom 4(n+1) Bell pairs (i.e., 8(n+1) photons). When successful, the(4(n+1)+n+2)-photon GHZ state includes a photon at each output channel320. The photon at each output channel 320 is entangled with the photonat each other output channel 320 of device 300-e such that the outputphotons are in a GHZ state (e.g., the set of output photons is in amaximally-entangled state).

Device 300-d is an example of device 300-e with n=1. Thus, FIG. 3Hillustrates that device 300-d can be extended to form an arbitrarilylarge GHZ state (e.g., a GHZ state having an arbitrarily large oddnumber of photons), albeit with decreasing probability of success as thesize of the GHZ state increases (e.g., because the outcomes measured byeach of the fusion gates must be indicative of success, which becomesless likely as there are more and more fusion gates).

FIGS. 4A-4B are schematic diagrams illustrating fusion gates, inaccordance with some embodiments. The fusion mechanisms provided bythese fusions gates allow for the construction of entangled qubit states(e.g., cluster states). These fusion gates have the advantage that theydo not require elaborate interferometers with multiple beam-splitters inseries. Further, these fusion gates make use of Hong-On-Mandel (HOM)effects and therefore require stability only over the coherence lengthof the photons rather than requiring phase stability of aMach-Zehnder-type interferometer (MZI), as some previous entanglementschemes have required.

FIGS. 4A-4B show Type-1 and Type-2 fusion gates in the polarizationencoding in accordance with some embodiments. However, one of ordinaryskill will appreciate that the disclosure is not limited to thepolarization encoding and any type of encoding can be employed, e.g.,path encoding, dual rail encoding, or any other encoding, withoutdeparting from the scope of the present disclosure. While one ofordinary skill will be familiar with the mapping between polarizationand path encoding, for convenience, FIG. 11 is provided herein to showan illustrative mapping between system components in thepolarization-encoded scheme and system components in the path-encodedscheme. Such a mapping can be applied to any of the polarization encodeddevices, systems, and methods described herein, not just the fusiongates described herein. Furthermore, in accordance with one or moreembodiments qubit states can be rotated using one or more Hadamard gateswhich can be implemented a number of different ways as described in FIG.9 . In the examples described below, Hadamard-like rotations on qubitsare described using the language of polarization encoding and are thuspolarization rotators but embodiments of the present disclosure are notlimited to merely a polarization encoding representation of the Hadamardgate.

FIG. 4A illustrates a photonic Type-I fusion gate 400-a (in thepolarization encoding), in accordance with some embodiments. In general,a Type-I fusion gate is a device that receives, as a first input atinput channel 402, a first qubit (e.g., a first photonic qubit) from afirst qubit cluster and receives, as a second input at second inputchannel 404, a second qubit from a second qubit cluster. Examples ofqubit clusters include, e.g., a multi-photon state that includesentanglement between two or more photonic qubits (such as a Bell stateor GHZ state, or larger entangled state), or a multi-qubit entangledstate comprising one or more matter-based qubits, e.g., ion and thelike, that are entangled with one or more photons or other matter-basedqubits. A Type-I fusion gate performs one or more operations on inputqubits of the input clusters to generate entanglement between thepreviously unentangled qubit clusters. Success of the fusion gate, i.e.,the creation of the desired entanglement between the input qubitclusters, is heralded by the detection of a certain number of qubits bythe detector 414. More specifically, a Type-I fusion gate, whensuccessful, measures a single measurement qubit at detector 414 andoutputs a single output qubit (at output channel 408) that has inheritedall of the entangling bonds of the input qubits. More specifically,following the measurement of the single measurement qubit (e.g., whensuccessful), the first qubit cluster and the second qubit cluster arerespectively entangled with the single output qubit (e.g., the singleoutput qubit is entangled with one or more qubits from the first qubitcluster and entangled with one or more qubits from the second qubitcluster, providing an entangling bond between the first qubit clusterand the second qubit cluster). Measurement of a single-qubit (e.g., onlyone photon) by the Type-I fusion gate heralds the success of the Type-Ifusion gate (e.g., indicates that the first qubit cluster and the secondqubit cluster are entangled).

Returning to FIGS. 4A, Type-I fusion gate 400-a is coupled with a firstinput channel 402 and a second input channel 404. The Type-I fusion gate400-a further includes a beam splitter 406 (e.g., a polarizing beamsplitter); a first output channel 408; a second output channel 410 whichpasses through a polarization rotator 412 (e.g., a 45-degreepolarization rotator); and a detector 414.

Type-I fusion gate 400-a receives a qubit (e.g., a first qubit) viafirst input channel 402 and a qubit (e.g., a second qubit) via secondinput channel 404. The Type-I fusion gate 400-a swaps some of theconstituent modes of the input two qubits (e.g., the horizontallypolarized mode of the first qubit can be swapped with the horizontallypolarized mode of the second qubit thereby forming a new qubit) usingthe polarizing beam splitter (PBS) 406. Any qubits output on outputchannel 410 will pass through polarization rotator 412 and will havetheir polarizations rotated, e.g., by 45 degrees (thereby eliminatingthe possibility that a detection of a qubit at the detector 414 canprovide a determination that the qubit originated from input 402 or 404.The Type-I fusion gate 400-a measures any polarization rotated qubitsreceived from output channel 410 with a qubit counter. The Type-I fusiongate 400-a succeeds when one qubit (either horizontally or verticallypolarized) is detected (e.g., which happens 50% of the time) and failswhen zero or two qubits are detected.

When the Type-I fusion gate 400-a succeeds, the two input qubits aresaid to be “fused” into a single output qubit that inherits all thebonds from the input qubits. More precisely, the output qubit is formedfrom one mode from each of the individual input qubits but applicationof the polarization beam splitter and polarization rotators eliminatethe possibility of knowing which qubit contributed which mode, so forsimplicity we refer to the output qubit as a new “output qubit.” Whenthe Type-I fusion gate 400-a fails, e.g., when two qubits are detectedby the detector 414, no output qubit is be generated and the Type-Ifusion gate 400-a has the effect of measuring both input qubits in thecomputational basis.

FIG. 4B illustrates a Type-II fusion gate 400-b, in accordance with someembodiments. A Type-II fusion gate is a device that receives, as a firstinput, a first qubit from a first qubit cluster and receives, as asecond input, a second qubit from a second qubit cluster. The Type-IIfusion gate makes use of redundant encoding, whereby a single logicalqubit is represented by multiple constituent qubits (thus measurement,in the computational basis, of the qubits provided the Type-II fusiongate does not destroy the encoding of the qubit).

The first input qubit (input on input channel 416) can be entangled witha respective qubit from the first qubit cluster and the second inputqubit (in put on input channel 418) can be entangled with a respectivequbit from the second qubit cluster. A Type-II fusion gate performs oneor more operations on the input qubits (e.g., swaps modes the twoqubits). A Type-II fusion gate includes two distinct detectors that(when successful) each measures a single-qubit that is provided to theType-II fusion gate. Following the measurement of the both qubits, whensuccessful, the first qubit cluster and the second qubit cluster areentangled through the respective qubits from the first qubit cluster andthe second qubit cluster. For example, the respective qubit from thefirst qubit cluster is entangled with one or more other qubits from thefirst qubit cluster and the respective qubit from the second qubitcluster is entangled with one or more qubits from the second qubitcluster. Since the respective qubit from the first qubit cluster is alsoentangled with the respective qubit from the second qubit cluster, therespective qubits provide a “bond” between the first qubit cluster andthe second qubit cluster).

Measuring one (and only one) qubit at each detector heralds the successof the Type-II fusion gate (e.g., the measurement indicates that thefirst qubit cluster and the second qubit cluster are entangled, or“fused”). Thus, since two qubits are provided to the Type-II fusiongate, and each qubit is detected by a distinct detector when successful,the Type-II fusion gate has the additional advantage that it does notrequire qubit-number discriminating detectors (e.g., detection of zeroqubits at one detector implies detection of two qubits at the other, sothere is no need to distinguish between detector of one qubit anddetection of two qubits).

To that end, the Type-II fusion gate 400-b is coupled with a first inputchannel 416 of the Type-II fusion gate 400-b and a second input channel418 of the Type-II fusion gate 400-b. The Type-II fusion gate 400-bfurther includes polarization rotators 419 and 420, corresponding tofirst input channel 416 and second input channel 418, respectively.

The Type-II fusion gate 400-b further includes a beam splitter 422(e.g., a polarizing beam splitter); a first output channel 424 whichpasses through a polarization rotator 426 (e.g., a 45-degreepolarization rotator); a second output channel 428 which passes througha polarization rotator 430 (e.g., a 45 degree polarization rotator); anddetectors 432 and 434, corresponding to first output channel 424 andsecond output channel 428, respectively.

The Type-II fusion gate 400-b receives a qubit (e.g., a first qubit) viafirst input channel 416 and a qubit (e.g., a second qubit) via secondinput channel 418. Type-II fusion gate 400-b rotates the state (e.g.,via a polarization rotation) of the qubits using polarization rotators419 and 420, mixes the qubits using polarizing beam splitter 422,rotates the polarization of the qubits again using polarization rotators426 and 430, and then measures the qubits in the rotated basis. TheType-II fusion gate 400-b is successful when a single qubit is detectedat each detector 432 and 434. Its effect is to project a pair of logicalqubits into a maximally-entangled state. When the gate fails, theType-II fusion gate 400-b measures zero or two qubits in one of thedetectors 432 and 434 (thus, no qubit detection in either of detector432 or 434 is indicative of failure). The effect of a failure is toperform a measurement in a computational basis of each of the qubits,removing them from the redundant encoding but not destroying the logicalqubit.

FIG. 5 is a schematic diagram illustrating an integrated photonics modecoupler 500 (e.g., a directional coupler also referred to herein as abeamsplitter), in accordance with some embodiments. Mode coupler 500includes a first channel 502-a and a second channel 502-b (e.g., formedphotolithographically as waveguides on a silica chip). Thus, in someembodiments, mode coupler 500 (as well as other optical componentsdescribed herein) is implemented using integrated optics. In contrast tobulk optics, which are inherently non-scalable and unreliable on a largescale, integrated optics provide stability and control over the opticalpath length, with the added advantage that the device size isdramatically reduced as compared to analogous circuits constructed usingbulk optics.

Mode coupler 500 is realized by positioning a portion of first channel(mode) 502-a and second channel (mode) 502-b closely enough so that theevanescent field of a photon in one mode couples to the other mode (andvice-versa). By controlling the separation between the waveguides(modes) and/or the length of the coupling region, different split ratioscan be obtained.

As shown in FIG. 11 , the photonic mode coupler shown in FIG. 5 isequivalent to a polarization rotator in the polarization encodingdevices and systems. As used herein, structures like the mode coupler500 are referred to as beamsplitters when implemented in a path encodeddevice. Likewise, as used herein, the term beamsplitter also includesthe polarization beamsplitters described in the context of thepolarization encoding. Thus, the use of the term “beamsplitter” hereinincludes a broad class of components that includes components thatprovide mode swap functionality (as in the case for the polarizationbeam splitter in the polarization encoding) and also includes modecoupling functionality (as in the case of a photonic beam splitter inthe path encoding). The correspondence between the components in thedifferent encodings is well understood in the art and is furthersummarized in FIG. 11 .

FIG. 6 is a schematic diagram illustrating a device 600 for generatingBell pairs (i.e., maximally entangled states of two qubits), inaccordance with some embodiments. For simplicity, aspects of FIG. 6described elsewhere in this document are not described again here.Device 600 is represented here as a device that employs photonic qubitsin the polarization encoding but one or ordinary skill will recognizethat any encoding can be used, e.g., path encoding, without departingfrom the scope of the present disclosure. Device 600 receives aplurality of photons 604 (e.g., four photons 604-a through 604-d). Insome embodiments, the photons 604 are received from single photonsources. Photons 604-a and 604-b are mixed using a beam splitter 606,with appropriate polarization rotation. Photons 604-c and 604-d aremixed using a beam splitter 608, with appropriate polarization rotation.A photon 610-a is output from beam splitter 606 and a photon 610-b isoutput from beam splitter 608. Another photon from each of beam splitter606 and beam splitter 608 is applied to a Type-II fusion gate 400-b, asdescribed above. The Type-II fusion gate 400-b determines whether photon610-a and photon 610-b are entangled in a Bell state. Thus, whensuccessful, photon 610-a and photon 610-b can be output as an entangledBell state (e.g., photon 610-a can be used as a first photon in FIGS.3A-3H and photon 610-b can be used as a second photon in FIGS. 3A-3H).

FIGS. 7A-7C illustrate a flow chart for a method 700 of generatingentangled qubit states, in accordance with some embodiments. In someembodiments, the entangled qubit states are maximally entangled (e.g.,GHZ states). In some embodiments, the entangled qubit states are used asresources in a quantum computing architecture (e.g., as an algorithmicresource or a precursor resource from which larger resources are builtup).

The method 700 includes receiving (702) a plurality of photon pairs(e.g., photon pairs 308, FIGS. 3A-3H). Each photon pair is in a Bellstate and includes a first photon and a second photon that is distinctand separate from the first photon. A Bell pair is a pair of qubits in aBell state (e.g., any Bell state). In some embodiments, the Bell statesare the maximally-entangled quantum states of a pair of qubits.

In some embodiments, each photon pair is a two-photonpolarization-entangled pair (e.g., each photon represents a qubitencoded in the photon's polarization degree of freedom). For example,the computational basis for the qubit is a vertical polarization v ofthe photon and a horizontal polarization h of the photon, and the twophotons in the Bell pair are entangled in this computational basis. Inthis basis, for two photons, there are four Bell states:

$\begin{matrix}{\left. \left| \Phi^{+} \right. \right\rangle = \frac{\left. \left. {\left. \left. \left| h \right. \right\rangle \middle| h \right\rangle +} \middle| v \right\rangle \middle| v \right\rangle}{\sqrt{2}}} & (10)\end{matrix}$ $\begin{matrix}{\left. \left| \Phi^{-} \right. \right\rangle = \frac{\left. \left. {\left. \left. \left| h \right. \right\rangle \middle| h \right\rangle -} \middle| v \right\rangle \middle| v \right\rangle}{\sqrt{2}}} & (11)\end{matrix}$ $\begin{matrix}{\left. \left| \Psi^{+} \right. \right\rangle = \frac{\left. \left. {\left. \left. \left| h \right. \right\rangle \middle| v \right\rangle +} \middle| v \right\rangle \middle| h \right\rangle}{\sqrt{2}}} & (12)\end{matrix}$ $\begin{matrix}{\left. \left| \Psi^{-} \right. \right\rangle = \frac{\left. \left. {\left. \left. \left| h \right. \right\rangle \middle| v \right\rangle -} \middle| v \right\rangle \middle| h \right\rangle}{\sqrt{2}}} & (13)\end{matrix}$

The method 700 includes obtaining a plurality of photons comprising atleast six photons in a Greenberger-Horn-Zeilinger (GHZ) state (e.g.,entangled in a polarization degree of freedom of the photons) byproviding photons of the plurality of photon pairs to a plurality ofbeam splitters (e.g., as described with reference to the operations 704,706, and optionally operations 714 and 722, described below). In someembodiments, the method 700 includes attempting to obtain a plurality ofphotons comprising at least six photons in a Greenberger-Horn-Zeilinger(GHZ) state, and the method 700 produces the at least six photons aswell as information indicating whether the GHZ state generation wassuccessful (e.g., whether the at least six photons are in aGreenberger-Horn-Zeilinger (GHZ) state). Thus, the at least six photonsin the GHZ state are heralded as being in the GHZ state.

In a computational basis with two states, a Greenberger-Horne-Zeilingerstate is a quantum superposition of all qubits being in a first state ofthe two states superposed with all of qubits being in a second state.Using photon polarization as the computational basis, the generalM-photon GHZ state can be written as:

$\begin{matrix}{\left. {❘{GHZ}} \right\rangle = \frac{\left. {{{\left. {❘h} \right\rangle^{\otimes M} +}❘}v} \right\rangle^{\otimes M}}{\sqrt{2}}} & (14)\end{matrix}$

The method 700 includes providing (704) a first photon of a first photonpair as a first input to a first beam splitter (e.g., via first inputchannel 304-a of first beam splitter 302-a, FIGS. 3A-3H) and a firstphoton of a second photon pair as a second input to the first beamsplitter (e.g., via second input channel 306-a of first beam splitter302-a, FIGS. 3A-3H). The first beam splitter is coupled with a firstoutput channel of the first beam splitter (e.g., first output channel310-a, FIGS. 3A-3H) and a second output channel of the first beamsplitter (e.g., second output channel 312-a, FIGS. 3A-3H). In someembodiments, the first beam splitter is a polarizing beam splitter (PBS)(e.g., as described with reference to FIG. 5 ). In some embodiments, apolarizing beam splitter operates on one or more photons in a firstchannel having a first polarization (e.g., a horizontal polarization) bymaintaining the one or more photons in the first channel (e.g.,transfers the one or more photons from an input channel to acorresponding output channel that is a waveguide continuation of theinput channel). In some embodiments, a polarizing beam splitter operateson one or more photons in a first channel having a second polarization(e.g., a vertical polarization) by transferring the one or more photonsin to a second channel distinct from the first channel (e.g., transfersthe one or more photons from an input channel to an output channel thatis not a waveguide continuation of the input channel).

Thus, in some embodiments, the operation of the polarizing beam splittercan be written as follows:S|h,0

_(i) =|h,0

_(o)  (15)S|v,0

_(i)=|0,v

_(o)  (16)S|0,h

_(i)=|0,h

_(o)  (17)S|0,v

_(i) =|v,0

_(o)  (18)

In Equations ((15)-(18) above, the operator S represents the operationof the polarizing beam splitter, the first value a in the ket |a, b

_(i) represents a polarization of a photon in the first input channel ofthe polarizing beam splitter (e.g., a=h represents ahorizontally-polarized photon in the first input channel of thepolarizing beam splitter, a=v represents a vertically-polarized photonin the first input channel of the polarizing beam splitter, and a=0represents zero photons in the first input channel of the polarizingbeam splitter), the second value b in the ket |a, b

_(i) represents a polarization of a photon in the first input channel ofthe polarizing beam splitter, the third value c in the ket |c, d

_(o) represents a polarization of a photon in the first output channelof the polarizing beam splitter, and the fourth value din the ket |c, d

_(o) represents a polarization of a photon in the second output channelof the polarizing beam splitter. Thus, as used herein, a comma within aket separates a photon's state with respect to plurality of channelswhereas multiple kets are used to signify different photons.

In some embodiments, the first photon from the first photon pair and thefirst photon from the second photon pair are provided in a firstpolarization (e.g., a horizontal polarization or a verticalpolarization). In some embodiments, the polarization of the first photonfrom the first photon pair and the first photon from the second photonpair are rotated (e.g., by 45 degrees) before entering the first beamsplitter (e.g., the first input channel of the first beam splitter andthe second input channel of the first beam splitter include or are actedupon by a polarization rotator). In some embodiments, the polarizationof the first photon from the first photon pair and the first photon fromthe second photon pair are rotated (e.g., by 45 degrees) after exitingthe first beam splitter (e.g., the first output channel of the firstbeam splitter and the second output channel of the first beam splitterinclude or are acted upon by a polarization rotator).

In some embodiments, the polarizations are rotated using a polarizationrotator. The operation of a 45-degree polarization rotator, for example,can be written as follows:

$\begin{matrix}{\left. {R{❘h}} \right\rangle = \frac{\left. {{{\left. {❘h} \right\rangle +}❘}v} \right\rangle}{\sqrt{2}}} & (19)\end{matrix}$ $\begin{matrix}{\left. {R{❘v}} \right\rangle = \frac{\left. {{{\left. {❘v} \right\rangle -}❘}h} \right\rangle}{\sqrt{2}}} & (20)\end{matrix}$

In Equations (19)-(20) above, the operator R represents the operation ofthe 45 degree polarization rotator.

The method 700 includes providing (706) a first photon of a third photonpair as a first input to a second beam splitter (e.g., via first inputchannel 304-b of second beam splitter 302-b, FIGS. 3A-H) and a firstphoton of a fourth photon pair as a second input to the second beamsplitter (e.g., via second input channel 306-b of second beam splitter302-b, FIGS. 3A-3H). The second beam splitter is distinct from the firstbeam splitter. In some embodiments, the second beam splitter is apolarizing beam splitter. The second beam splitter is coupled with afirst output channel of the second beam splitter (e.g., first outputchannel 310-b, FIGS. 3A-3H) and a second output channel of the secondbeam splitter (e.g., second output channel 312-b, FIGS. 3A-3H). In someembodiments, the polarization of the first photon from the third photonpair and the first photon from the fourth photon pair are rotated (e.g.,by 45 degrees) before entering the second beam splitter (e.g., the firstinput channel of the second beam splitter and the second input channelof the second beam splitter include or are acted upon by a polarizationrotator). In some embodiments, the polarization of the first photon fromthe third photon pair and the first photon from the fourth photon pairare rotated (e.g., by 45 degrees) after exiting the second beam splitter(e.g., the first output channel of the second beam splitter and thesecond output channel of the second beam splitter include or are actedupon by a polarization rotator).

The method 700 includes providing (708) a photon output from the firstbeam splitter in the second output channel of the first beam splitter asa first input to a first fusion gate (e.g., first fusion gate 314-a,FIGS. 3A-3H) and a photon output from the second beam splitter in thefirst output channel of the second beam splitter as a second input tothe first fusion gate.

In some embodiments, method 700 includes determining whether successcriteria are met (e.g., GHZ generation success criteria indicating thatthe desired entangled state has been generated). In some embodiments,determining whether the success criteria are met includes detecting astate of one or more photons provided to one or more fusion gates (e.g.,where the photon's state includes its presence in a particular channeland/or the photon's polarization). In some embodiments, the method 700includes detecting (710), using the first fusion gate, a first state ofthe photon provided as the first input to the first fusion gate and asecond state of the photon provided as the second input to the firstfusion gate (e.g., the first fusion gate is a Type-II fusion gate). Thefirst state of the photon provided as the first input to the firstfusion gate and the second state of the photon provided as the secondinput to the first fusion gate herald the success or failure of thefirst fusion gate's operation. In some embodiments (e.g., when thedevice includes a plurality of fusion gates, the success criteriainclude detection of only one photon at each detector of each fusiongate of the device).

In some embodiments, detecting the first state includes detecting aphoton in a first output channel of the first fusion gate (e.g., thestate is or includes the photon's presence in the first output channel)and detecting the second state includes detecting a photon in a secondoutput channel of the first fusion gate (e.g., the state is or includesthe photon's presence in the first output channel). To that end, in someembodiments, the first fusion gate includes a first detector and asecond detector that is distinct and separate from the first detector.Detecting operation 710 includes (e.g., when successful): detecting,using the first detector of the first fusion gate, only one photonselected from the photon provided as the first input to the first fusiongate and the photon provided as the second input to the first fusiongate; and detecting, using the second detector of the first fusion gate,only another photon selected from the photon provided as the first inputto the first fusion gate and the photon provided as the second input tothe first fusion gate. In some embodiments, when the first detectordetects both photons (e.g., and the second detector detects zerophotons) or the second detector detects both photons (e.g., and thefirst detector detects zero photons), the method 700 includesdetermining that the attempt to prepare the GHZ state has failed (e.g.,the success criteria are not met).

For example, in some embodiments, the GHZ state is a 6-photon GHZ stateproduced using 4 pairs of photons in a Bell state. In some embodiments,the method 700 includes determining (712) whether the first state andthe second state are indicative of a 6-photon GHZ state. The remainingphotons comprise the 6 photons in the 6-photon GHZ state. For example,the plurality of photons in the GHZ state includes: second photons of:the first photon pair, the second photon pair, the third photon pair,and the fourth photon pair; one photon output from the first beamsplitter in the first output channel of the first beam splitter; and onephoton output from the second beam splitter in the second output channelof the second beam splitter.

In some embodiments, the method 700 includes providing (714) a firstphoton of a fifth photon pair as a first input to a third beam splitter(e.g., via first input channel 304-c of third beam splitter 302-c, FIGS.3C-3H) and a first photon of a sixth photon pair as a second input tothe third beam splitter (e.g., via second input channel 306-c of thirdbeam splitter 302-c, FIGS. 3C-3H). The third beam splitter is distinctfrom the first beam splitter and the second beam splitter. The thirdbeam splitter is coupled with a first output channel of the third beamsplitter (e.g., first output channel 310-c, FIGS. 3C-3H) and a secondoutput channel of the third beam splitter (e.g., second output channel312-c, FIGS. 3C-3H). In some embodiments, the polarization of the firstphoton from the fifth photon pair and the first photon from the sixthphoton pair are rotated (e.g., by 45 degrees) before entering the thirdbeam splitter (e.g., the first input channel of the third beam splitterand the second input channel of the third beam splitter include or areacted upon by a polarization rotator). In some embodiments, thepolarization of the first photon from the fifth photon pair and thefirst photon from the sixth photon pair are rotated (e.g., by 45degrees) after exiting the third beam splitter (e.g., the first outputchannel of the third beam splitter and the second output channel of thethird beam splitter include or are acted upon by a polarizationrotator).

In some embodiments, the method 700 includes providing (716) a photonoutput from the second beam splitter in the second output channel of thesecond beam splitter as a first input to a second fusion gate (e.g.,second fusion gate 314-b, FIGS. 3C-3D) and a photon output from thethird beam splitter in the first output channel of the third beamsplitter as a second input to the second fusion gate.

In some embodiments, the second fusion is a Type-II fusion gate. In someembodiments, the method 700 includes detecting (718), using the secondfusion gate, a third state of the photon provided as the first input tothe second fusion gate and a fourth state of the photon provided as thesecond input to the second fusion gate (e.g., using detectors as shownin FIG. 6 ). The third state of the photon provided as the first inputto the second fusion gate and the fourth state of the photon provided asthe second input to the second fusion gate herald the success or failureof the second fusion gate's operation.

For example, in some embodiments, the GHZ state is an 8-photon GHZ stateproduced using 6 pairs of photons in a Bell state. In some embodiments,the method 700 includes determining (720) whether the first state, thesecond state, the third state, and the fourth state are indicative of an8-photon GHZ state. The remaining photons comprise the 8 photons in the8-photon GHZ state. For example, the plurality of photons in the GHZstate includes: second photons of: the first photon pair, the secondphoton pair, the third photon pair, the fourth photon pair, the fifthphoton pair, and the sixth photon pair; one photon output from the firstbeam splitter in the first output channel of the first beam splitter;and one photon output from the third beam splitter in the second outputchannel of the third beam splitter.

In some embodiments, the method 700 includes providing (722) a firstphoton of a seventh photon pair as a first input to a fourth beamsplitter (e.g., via first input channel 304-d of fourth beam splitter302-d, FIG. 3F-3H) and a first photon of an eighth photon pair as asecond input to the fourth beam splitter (e.g., via second input channel306-d of fourth beam splitter 302-d, FIG. 3F-3H). The fourth beamsplitter is distinct from the first beam splitter, the second beamsplitter, and the third beam splitter. The fourth beam splitter iscoupled with a first output channel of the fourth beam splitter (e.g.,first output channel 310-d, FIGS. 3F-3H) and a second output channel ofthe fourth beam splitter (second output channel 312-d, Figures, 3F-3H).In some embodiments, the polarization of the first photon from theseventh photon pair and the first photon from the eighth photon pair arerotated (e.g., by 45 degrees) before entering the fourth beam splitter(e.g., the first input channel of the fourth beam splitter and thesecond input channel of the fourth beam splitter include or are actedupon by a polarization rotator). In some embodiments, the polarizationof the first photon from the seventh photon pair and the first photonfrom the eighth photon pair are rotated (e.g., by 45 degrees) afterexiting the fourth beam splitter (e.g., the first output channel of thefourth beam splitter and the second output channel of the fourth beamsplitter include or are acted upon by a polarization rotator).

In some embodiments, the method 700 includes providing (724) a photonoutput from the third beam splitter in the second output channel of thethird beam splitter as a first input to a third fusion gate (e.g.,second fusion gate 314-c, FIGS. 3F-3H) and a photon output from thefourth beam splitter in the first output channel of the fourth beamsplitter as a second input to the third fusion gate.

In some embodiments, the second fusion gate and the third fusion gateare Type-II fusion gates. In some embodiments, the GHZ state is a10-photon GHZ state produced using 8 pairs of photons in a Bell state.In some embodiments, the second fusion gate is a Type-I fusion gate andthe third fusion gate is a Type-II fusion gate. In some embodiments, theGHZ state is an 11-photon GHZ state produced using 8 pairs of photons ina Bell state.

More generally, the GHZ state is an n-photon GHZ state. The first fusiongate is a fusion gate in a plurality of fusion gates (e.g., thatincludes the second fusion gate and the third fusion gate, describedabove). The method 700 further includes detecting, using the pluralityfusion gates, a state of photons provided to the plurality of fusiongates and determining whether the state of the photons provided to theplurality of fusion gates is indicative of an n-photon GHZ state. Insome embodiments, the state of the photons provided to the plurality offusion gates is indicative of successful generation of the n-photon GHZstate when all of the fusion gates indicate success (e.g., in ananalogous manner to the success criteria described above with referenceto operation 710).

FIG. 8 is a schematic diagram illustrating an architecture of a device800 for generating entangled qubit states, e.g., photonic qubit states,in accordance with some embodiments. In some embodiments, a plurality ofsingle photon sources 802 provides single photons to a plurality of Bellpair generators 600. In some embodiments, the single photon sources 802and/or the Bell pair generators 600 are part of device 800.Alternatively, the single photon sources 802 and/or the Bell pairgenerators 600 are external to device 800.

For example, four single photon sources 802-a through 802-d providesingle photons to Bell pair generator 600-a; four single photon sources802-e through 802-h provide single photons to Bell pair generator 600-b;four single photon sources 802-i through 802-1 provide single photons toBell pair generator 600-c; and four single photon sources 802-m through802-p provide single photons to Bell pair generator 600-d. Bell pairgenerators 600 generate Bell pairs (e.g., as described with reference toFIG. 6 ), and in doing so consume two of the four photons provided toeach Bell pair generator 600. Thus, each Bell pair generator 600outputs, when successful, two photons in an entangled Bell state. TheBell pairs from each Bell pair generator is provided to device 300-a, asdescribed with reference to FIGS. 3A-3B, which generates a 6-photon GHZstate from the 4 Bell pairs provided.

It will be apparent to one of skill in the art how to modify device 800(e.g., with additional photon sources 802 and Bell pair generators) toincorporate other devices 300 besides device 300-a (e.g., any of device300-b through device 300-e, FIGS. 3C-3H).

IV. Notation

The schematic diagrams used herein illustrate certain components/quantumgates. FIG. 9 illustrate a simplified notation for some of thesecomponents/quantum gates (in particular, beamsplitters and n-modeHadamards). In general, the definitions and relations betweenbeamsplitters and n-mode Hadamards can be translated to the path-encodeddiagrams using the notation shown in FIG. 9 . Mathematically, theimaginary Hadamard can be written as

$h^{i} = {\frac{1}{\sqrt{2}}\begin{pmatrix}1 & i \\i & 1\end{pmatrix}}$and the real Hadamard h^(r) can be written as

$h^{r} = {\frac{1}{\sqrt{2}}\begin{pmatrix}1 & 1 \\1 & {- 1}\end{pmatrix}}$Physically, e.g., in a photonic system, the above Hadamard gates can beimplemented as beamsplitters and/or directional couplers. The real andcomplex Hadamards can be transformed into one another by applying a ±iphase shift to the second mode. The unitary operators that define such aphase shift are given by

$s = \begin{pmatrix}1 & 0 \\0 & i\end{pmatrix}$and

${s^{\dagger} = \begin{pmatrix}1 & 0 \\0 & {- i}\end{pmatrix}},$in which case h^(i)=sh^(r)s and h^(r)=s^(†)h^(i)s^(†).

In view of the above mathematical relations, the complex Hadamardcorresponds to a real Hadamard preceded and followed by a phase of i onthe second mode, and the real Hadamard corresponds to a complex Hadamardpreceded and followed by a phase of −i on the second mode. Both matricesare symmetric, but they differ in that h^(i) applies the same operationto both the modes it acts on, while h^(r) acts differently on the twomodes. This means that, while the order of the input modes is irrelevantwhen the complex Hadamard is used, it is important if the real versionis applied.

The two-mode imaginary Hadamard h^(i) and the two-mode real Hadamardh^(r) can be represented schematically as mode couplers 903 and 905,respectively. The transformations between the two are also shown viaschematic elements 907, where −i phase shifts applied to a mode arerepresented by open boxes and i phase shifts applied to a mode arerepresented by boxes with black fill. As already described above, thesemode couplers can be physically implemented as beamsplitters,directional couplers and the like.

The above description for two-mode Hadamard gates can be generalized ton-mode situations. More specifically an n-mode (also referred to hereinas an n-th order Hadamard) real/imaginary Hadamard can be expressed asH _(n) ^(r) =h ^(r) ⊗h ^(r) ⊗ . . . ⊗h ^(r) =h ^(r⊗n)H _(n) ^(i) =h ^(i) ⊗h ^(i) ⊗ . . . ⊗h ^(i) =h ^(i⊗n)For example, the 2^(nd) order Hadamards are

$\begin{matrix}{H_{2}^{r} = {\frac{1}{2}\begin{pmatrix}1 & 1 & 1 & 1 \\1 & {- 1} & 1 & {- 1} \\1 & 1 & {- 1} & {- 1} \\1 & {- 1} & {- 1} & 1\end{pmatrix}}} \\{H_{2}^{i} = {\frac{1}{2}\begin{pmatrix}1 & i & i & {- 1} \\i & 1 & {- 1} & i \\i & {- 1} & 1 & i \\{- 1} & i & i & 1\end{pmatrix}}}\end{matrix}$More generally, the 2n×2n Hadamards (real or complex) can be decomposedinto products of single beamsplitters using the following formula:

$H_{n}^{r(i)} = {\prod\limits_{j = 1}^{\log_{2}(N)}\left( {\prod\limits_{i = {{0i{mod}\ 2^{j}} < 2^{j - 1}}}^{N - 1}h_{i,{i + 2^{j - 1}}}^{r(i)}} \right)}$Where N=2^(n) and the lower indices on h^(r(i)) indicate the modes thebeamsplitters act on. For example, expanding this formula gives:H ₁ ^(r(i)) =h _(0,1) ^(r(i))  (2×2)H ₂ ^(r(i)) =h _(0,1) ^(r(i)) h _(2,3) ^(r(i)) h _(0,2) ^(r(i)) h _(1,3)^(r(i))  (4×4)H ₃ ^(r(i)) =h _(0,1) ^(r(i)) h _(2,3) ^(r(i)) h _(4,5) ^(r(i)) h _(6,7)^(r(i)) h _(0,2) ^(r(i)) h _(1,3) ^(r(i)) h _(4,6) ^(r(i)) h _(5,7)^(r(i)) h _(0,4) ^(r(i)) h _(1,5) ^(r(i)) h _(2,6) ^(r(i)) h _(3,7)^(r(i))  (8×8)Schematic diagrams 909 show one example of the real second orderHadamard. Likewise, schematic 911 shows the imaginary second orderHadamard. Also included are the steps by which the real Hadamard can beconverted to the imaginary Hadamard.

V. Exemplary Beamsplitters

FIGS. 10A-10C illustrate schematic diagrams of waveguide beam splitters1000 (e.g., 1000 a, 1000 b, and 1000 c, respectively), in accordancewith some embodiments. In some embodiments, beam splitters areimplemented in integrated photonics via directional couplings, which arerealized by bringing together the different waveguides (e.g., waveguides1012 a and 1012 b) close enough so that the evanescent field of one ofthem can couple into the other one. By controlling the separation dbetween the waveguides 1012 and/or the length l of the coupling region1014, different transmissivity can be obtained and therefore thisimplementation is equivalent to a beam-splitter in bulk optics. In thismanner, waveguide beam splitter 1000 may be configured to have atransmissivity equal to 0.5 (e.g., a 50:50 beam splitter, greater than0.6, greater than 0.7, greater than 0.8, or greater than 0.9).

In some embodiments, waveguide beam splitters 1000 include variablephase-shifters 1016. Variable phase-shifters can be implemented inintegrated circuits, providing control over the relative phases of thestate of a photon spread over multiple modes. For the silica-on-siliconmaterials some embodiments implement variable phase-shifters usingthermo-optical switches. The thermo-optical switches use resistiveelements fabricated on the surface of the chip, that via thethermo-optical effect can provide a change of the refractive index n byraising the temperature of the waveguide 1012 by an amount of the orderof 10⁻⁵ K. One of skill in the art, however, having had the benefit ofthis disclosure, will understand that any effect that changes therefractive index of a portion of the waveguide can be used to generate avariable, electrically tunable, phase shift. For example, someembodiments use beam splitters based on any material that supports anelectro-optic effect, so-called χ² and χ³ materials such as lithiumniobite, BBO, KTP, and the like and even doped semiconductors such assilicon, germanium, and the like.

Beam-splitters with variable transmissivity and arbitrary phaserelationships between output modes can also be achieved by combiningdirectional couplings and variable phase-shifters in a Mach-ZehnderInterferometer (MZI) configuration, e.g., as shown in FIG. 10B. Completecontrol over the relative phase and amplitude of the two paths in dualrail encoding can be achieved by varying the phases imparted by phaseshifters 1016 a, 1016 b, and 1016 c. FIG. 10C shows a slightly simplerexample of a MZI that allows for a variable transmissivity between modes1012 a and 1012 b by varying the phase imparted by the phase shifter1016. FIGS. 10A-10C are only three examples of how one could implement amode coupling in a physical device, but any type of mode coupling/beamsplitter can be used without departing from the scope of the presentdisclosure.

For example, the waveguide beam splitter in FIG. 10C can be used toswitch photons in waveguide 1012 a into either waveguide 1012 a or 1012b by adjusting the phase of phase shifter 1016 b appropriately. Thus, atunable waveguide beam splitter is a device for mode swapping andoptical switching. In addition, these beam splitters, e.g., in a 50:50configuration can be used to spread the quantum state of a single photonequally across multiple modes (waveguides).

FIG. 12 shows one example of a path encoded system 1200 that correspondsto the system and device described in FIG. 3 . The system can beanalogously used to generate GHZ states as follows. Advantageously, inthe system shown in FIG. 12 , for every four Bell states received by theset of 16 qubit inputs, the six qubits in the GHZ state are output witha success probability that is larger than the previous record of 6.25%and more specifically, assuming an ideal system can be as high as18.75%.

The 16 qubit inputs are shown as inputs (also referred to herein as modepairs) 308-a 1, 308-a 2, 308-b 1, 308-b 2, 308-c 1, 308-c 2, 308-d 1,308-d 2. In the photonic case, the modes can be waveguides alreadydescribed above. To generate the GHZ state, four (dual rail pathencoded) qubit Bell pairs are received in these 18 modes, the photons ofeach Bell pair occupying a respective group of four of the 16 qubitinputs. For example, one Bell pair is input on pair of inputs 308-a 2and pair of inputs 308 a 1.

Portions of these Bell pairs are then sent to the interferometer system1210 that include a number of mode-couplers, also referred to herein abeamsplitters. These mode couplers can be arranged in many differentforms other than the one example shown here. More generally, any set ofmode-couplers can be used that takes any input mode and spreads itevenly amongst the various output modes. In this case the set ofbeamsplitters implements (up to a global phase) the 2-qubit imaginaryHadamard described and shown in FIG. 9 .

Returning back to the original 16 modes making up the 8 Bell pairs, 4mode swaps are implemented as shown such that one of the modes form eachbell pair are sent into the interferometer system 1120. For example, inFIG. 12 , modes 308-a 1, 308-b 1, 308-c 1, and 308-d 1 from the first,second, third and fourth Bell pairs are sent to the interferometersystem 1210.

As already described above, and reiterated below in reference to Table1, the device successfully generates a 6-GHZ state when at least twophotons are detected in no less than two of the four coupled modes.

More specifically a method for using the device of FIG. 12 forgenerating a 6-GHZ state include receiving a plurality of qubit pairs,each qubit pair being in a Bell state and including a first qubit and asecond qubit that is distinct and separate from the first qubit. Themethod further includes providing qubits of the plurality of qubit pairsto a plurality of beamsplitters, e.g., beamsplitters 1213 a, 1213 b,1213 c, 1213 d.

More specifically the method further includes providing a first mode ofa first qubit pair (e.g., mode 308-a 1) as a first input to a first beamsplitter (e.g., beamsplitter 1213 b) and a first mode of a second qubit(e.g., mode 308-b 1) pair as a second input to the first beam splitter,wherein the first beam splitter is coupled with a first output channel(the output channel coupled to detector 1212 a) of the first beamsplitter and a second output channel (the output channel coupled todetector 1212 b) of the first beam splitter.

The method further includes providing a first mode of a third qubit pair(e.g., mode 308-c 1) as a first input to a second beam splitter and afirst mode of a fourth qubit pair (e.g., 308-d 1) as a second input tothe second beam splitter (e.g., beamsplitter 1213 a) that is distinctfrom the first beam splitter. In some embodiments, the second beamsplitter is coupled with a first output channel of the second beamsplitter (e.g., the output channel coupled to detector 1212 c) and asecond output channel of the second beam splitter (e.g., the outputchannel coupled to the detector 1212 d).

The method further includes providing a qubit output from the first beamsplitter in the second output channel of the first beam splitter as afirst input to a first detector and a qubit output from the second beamsplitter in the first output channel of the second beam splitter as asecond input to the first detector. More specifically each of the qubitsthat are output from the first and second beamsplitters can be furtherequally spread amongst all four modes of the interferometer such thatthe probability that any given qubit is output to any of the fourdetectors 1212 a, 1212 b, 1212 c, 1212 d is equal.

Another equivalent way to describe the interferometer system 1210 is asa quantum information (or quantum state scrambling coupler) that erasesat its output any information regarding which modes the qubitsoriginated from. Each beamsplitter within the interferometer system 1210can be a 50/50 beam splitter, like that shown in FIG. 5 . The effect ofthe interferometer system 1210 is that detectors 1212 can detect aphoton but cannot identify which waveguide it came from, which in somecircumstances is necessary to generate the entanglement of the GHZstate.

In some embodiments, detectors 1212 are coupled to a digital logicmodule 1211 (e.g., which may be implemented as field programmabledigital logic using, for example, a field programmable gate array (FPGA)or an on-chip hard-wired circuit, such as an application specificintegrated circuit (ASIC)). Alternatively, in some embodiments, thedetectors 1212 are coupled to an off-chip classical computer (e.g.,classical computer 112, FIG. 1 ). In some embodiments, the digital logicmodule 1211 and/or the classical computer receives information from eachdetector 1212 indicating whether the detector 1212 detected a photon(and optionally how many). Stated another way, the digital logic module1211 and/or the classical computer receives the detection pattern for adetection operation from the detectors 1212 (e.g., in the form of analogdetection signals). The digital logic module 1211 and/or the classicalcomputer executes logic that configures a switch (not shown) to eitheroutput the photons, pass the photons to a subsequent stage of thedevice. In some embodiments, the digital logic module 1211 and/or theclassical computer does so by referencing a look-up table (e.g., storedin the memory) to determine whether the detection pattern indicates thatthe photons remaining in the first set of output waveguides 320 are inthe desired GHZ state of if a fusion failure has occurred.

Table 1 below provides an example of the logic performed by the digitallogic module 1211 and/or the classical computer. In table 1, a checkmark (✓) indicates successful generation of a GHZ state, an “X”indicates a failure. Further, in Table 1, the detection patterns arewritten, e.g., 1-0-1-0, which means that one photon is detected by afirst detector (e.g., detector 1212 a); zero photons are detected by asecond detector (e.g., detector 1212 b); one photon is detected by athird detector (e.g., detector 1212 c); and zero photons are detected bya fourth detector (e.g., detector 1212 d). “N/A” is used to signify thatthe stage is unnecessary, and therefore no detection pattern or outcomeis obtained for that stage.

TABLE 1 First Stage Detection Pattern GHZ Outcome 1-1-0-0 ✓ 1-0-1-0 ✓0-0-1-1 ✓ 0-1-0-1 ✓ 1-0-0-1 ✓ 0-1-1-0 ✓ 2-0-0-0 X

FIG. 12 is shown as merely one example of how to implement the systemsand methods described herein to generate entangled qubits states in thepath encoding, specifically the device shown in FIG. 3 that generates a6 GHZ state from 4 Bell pairs. One of ordinary skill having the benefitof this disclosure will recognize that any of the devices describedherein can be similarly implemented in the path encoding.

For the sake of conciseness, the qubits disclosed herein have beendescribed in the context of photonics, i.e., photonic qubits, and can beformed from one or more photons, but other types of qubits are alsopossible without departing from the scope of the present disclosure. Forexample, the qubits disclosed herein can be a collection of quantumsystems and/or particles and can be formed using any qubit architecture,such as massive particles such as atoms, ions, and/or nuclei. In otherexamples, the quantum systems can be other engineered quantum systemssuch as flux qubits, phase qubits, or charge qubits (e.g., formed from asuperconducting Josephson junction), topological qubits (e.g., Majoranafermions), or spin qubits formed from vacancy centers (e.g., nitrogenvacancies in diamond). Furthermore, for the sake of clarity ofdescription, the term “qubit” is used herein although the system canalso employ quantum information carriers that encode information in amanner that is not necessarily associated with a binary bit. Forexample, qudits can be used, i.e., quantum systems that can encodeinformation in more than two quantum states in accordance with someembodiments.

The terminology used in the description of the various describedembodiments herein is for the purpose of describing particularembodiments only and is not intended to be limiting. As used in thedescription of the various described embodiments and the appendedclaims, the singular forms “a”, “an” and “the” are intended to includethe plural forms as well, unless the context clearly indicatesotherwise. It will also be understood that the term “and/or” as usedherein refers to and encompasses any and all possible combinations ofone or more of the associated listed items. It will be furtherunderstood that the terms “includes,” “including,” “comprises,” and/or“comprising,” when used in this specification, specify the presence ofstated features, integers, steps, operations, elements, and/orcomponents, but do not preclude the presence or addition of one or moreother features, integers, steps, operations, elements, components,and/or groups thereof.

It will also be understood that, although the terms first, second, etc.,are, in some instances, used herein to describe various elements, theseelements should not be limited by these terms. These terms are only usedto distinguish one element from another. For example, a first beamsplitter could be termed a second beam splitter, and, similarly, asecond beam splitter could be termed a first beam splitter, withoutdeparting from the scope of the various described embodiments. The firstbeam splitter and the second beam splitter are both beam splitters, butthey are not the same beam splitter unless explicitly stated as such.

As used herein, the term “if” is, optionally, construed to mean “when”or “upon” or “in response to determining” or “in response to detecting”or “in accordance with a determination that,” depending on the context.

The foregoing description, for purpose of explanation, has beendescribed with reference to specific embodiments. However, theillustrative discussions above are not intended to be exhaustive or tolimit the scope of the claims to the precise forms disclosed. Manymodifications and variations are possible in view of the aboveteachings. The embodiments were chosen in order to best explain theprinciples underlying the claims and their practical applications, tothereby enable others skilled in the art to best use the embodimentswith various modifications as are suited to the particular usescontemplated.

What is claimed is:
 1. A device, comprising: a first beam splittercoupled with: a first input channel for the first beam splitterconfigured for receiving a first photon of a photon pair from a firstphoton source of a plurality of photon sources; and a second inputchannel for the first beam splitter configured for receiving a firstphoton of a photon pair from a second photon source of the plurality ofphoton sources; a first output channel for the first beam splitter; anda second output channel for the first beam splitter; a second beamsplitter coupled with: a first input channel for the second beamsplitter configured for receiving a first photon of a photon pair from athird photon source of the plurality of photon sources; a second inputchannel for the second beam splitter configured for receiving a firstphoton of a photon pair from a fourth photon source of the plurality ofphoton sources; a first output channel for the second beam splitter; anda second output channel for the second beam splitter; a first fusiongate coupled with: a first input channel, for the first fusion gate,coupled with the second output channel of the first beam splitter; and asecond input channel, for the first fusion gate, coupled with the firstoutput channel of the second beam splitter; and respective inputchannels for receiving second photons from: the photon pair from thefirst photon source, the photon pair from the second photon source, thephoton pair from the third photon source, and the photon pair from thefourth photon source; wherein each respective input channel forreceiving a second photon is coupled with a respective output channel.2. The device of claim 1, wherein the first fusion gate includes: afirst output channel of the first fusion gate configured to receive afirst set of one or more photons from the first input channel of thefirst fusion gate and the second input channel of the first fusion gate;a second output channel of the first fusion gate configured to receive asecond set of one or more photons, distinct from the first set of one ormore photons, from the first input channel of the first fusion gate andthe second input channel of the first fusion gate; a first detector ofthe first fusion gate configured to detect the first set of one or morephotons; and a second detector of the first fusion gate configured todetect the second set of one or more photons.
 3. The device of claim 2,further including: a third beam splitter coupled with: a first inputchannel for the third beam splitter configured for receiving a firstphoton of a photon pair from a fifth photon source of the plurality ofphoton sources; and a second input channel for the third beam splitterconfigured for receiving a first photon of a photon pair from a sixthphoton source of the plurality of photon sources; a first output channelfor the third beam splitter; and a second output channel for the thirdbeam splitter; a second fusion gate coupled with: a first input channel,for the second fusion gate, coupled with the second output channel ofthe second beam splitter; and a second input channel, for the secondfusion gate, coupled with the first output channel of the third beamsplitter; and respective input channels for receiving second photonsfrom: the photon pair from the fifth photon source and the photon pairfrom the sixth photon source; wherein each respective input channel forreceiving a second photon is coupled with a respective output channel.4. The device of claim 3, wherein the second fusion gate includes: afirst output channel of the second fusion gate configured to receive athird set of one or more photons from the first input channel of thesecond fusion gate and the second input channel of the second fusiongate; a second output channel of the second fusion gate configured toreceive a fourth set of one or more photons, distinct from the third setof one or more photons, from the first input channel of the secondfusion gate and the second input channel of the second fusion gate; afirst detector of the second fusion gate configured to detect the thirdset of one or more photons; and a second detector of the second fusiongate configured to detect the second set of one or more photons.
 5. Thedevice of claim 4, further including: a fourth beam splitter coupledwith: a first input channel for the fourth beam splitter configured forreceiving a first photon of a photon pair from a seventh photon sourceof the plurality of photon sources; and a second input channel for thefourth beam splitter configured for receiving a first photon of a photonpair from an eighth photon source of the plurality of photon sources; afirst output channel for the fourth beam splitter; and a second outputchannel for the fourth beam splitter; a third fusion gate coupled with:a first input channel, for the third fusion gate, coupled with thesecond output channel of the third beam splitter; and a second inputchannel, for the third fusion gate, coupled with the first outputchannel of the fourth beam splitter; and respective input channels forreceiving second photons from: the photon pair from the seventh photonsource and the photon pair from the eighth photon source; wherein eachrespective input channel for receiving a second photon is coupled with arespective output channel.
 6. The device of claim 5, wherein the thirdfusion gate includes: a first output channel of the third fusion gateconfigured to receive a fifth set of one or more photons from the firstinput channel of the third fusion gate and the second input channel ofthe third fusion gate; a second output channel of the third fusion gateconfigured to receive a sixth set of one or more photons, distinct fromthe fifth set of one or more photons, from the first input channel ofthe third fusion gate and the second input channel of the third fusiongate; a first detector of the third fusion gate configured to detect thethird set of one or more photons; and a second detector of the thirdfusion gate configured to detect the second set of one or more photons.7. The device of claim 5, wherein the third fusion gate includes: afirst output channel of the third fusion gate configured to receive afifth set of one or more photons from the first input channel of thethird fusion gate and the second input channel of the third fusion gate;a second output channel of the third fusion gate configured to receive asixth set of one or more photons, distinct from the fifth set of one ormore photons, from the first input channel of the third fusion gate andthe second input channel of the third fusion gate; and only one detectorof the third fusion gate configured to detect the third set of one ormore photons; wherein the sixth set of one or more photons is outputfrom the third fusion gate along the second output channel.
 8. A device,comprising: a first beam splitter coupled with: a first input channelfor the first beam splitter configured for receiving a first qubit of aqubit pair from a first qubit source of a plurality of qubit sources;and a second input channel for the first beam splitter configured forreceiving a first qubit of a qubit pair from a second qubit source ofthe plurality of qubit sources; a first output channel for the firstbeam splitter; and a second output channel for the first beam splitter;a second beam splitter coupled with: a first input channel for thesecond beam splitter configured for receiving a first qubit of a qubitpair from a third qubit source of the plurality of qubit sources; asecond input channel for the second beam splitter configured forreceiving a first qubit of a qubit pair from a fourth qubit source ofthe plurality of qubit sources; a first output channel for the secondbeam splitter; and a second output channel for the second beam splitter;a first fusion gate coupled with: a first input channel, for the firstfusion gate, coupled with the second output channel of the first beamsplitter; and a second input channel, for the first fusion gate, coupledwith the first output channel of the second beam splitter; andrespective input channels for receiving second qubits from: the qubitpair from the first qubit source, the qubit pair from the second qubitsource, the qubit pair from the third qubit source, and the qubit pairfrom the fourth qubit source; wherein each respective input channel forreceiving a second qubit is coupled with a respective output channel. 9.The device of claim 8, wherein the first fusion gate includes: a firstoutput channel of the first fusion gate configured to receive a firstset of one or more qubits from the first input channel of the firstfusion gate and the second input channel of the first fusion gate; asecond output channel of the first fusion gate configured to receive asecond set of one or more qubits, distinct from the first set of one ormore qubits, from the first input channel of the first fusion gate andthe second input channel of the first fusion gate; a first detector ofthe first fusion gate configured to detect the first set of one or morequbits; and a second detector of the first fusion gate configured todetect the second set of one or more qubits.
 10. The device of claim 9,further including: a third beam splitter coupled with: a first inputchannel for the third beam splitter configured for receiving a firstqubit of a qubit pair from a fifth qubit source of the plurality ofqubit sources; and a second input channel for the third beam splitterconfigured for receiving a first qubit of a qubit pair from a sixthqubit source of the plurality of qubit sources; a first output channelfor the third beam splitter; and a second output channel for the thirdbeam splitter; a second fusion gate coupled with: a first input channel,for the second fusion gate, coupled with the second output channel ofthe second beam splitter; and a second input channel, for the secondfusion gate, coupled with the first output channel of the third beamsplitter; and respective input channels for receiving second qubitsfrom: the qubit pair from the fifth qubit source and the qubit pair fromthe sixth qubit source; wherein each respective input channel forreceiving a second qubit is coupled with a respective output channel.11. The device of claim 10, wherein the second fusion gate includes: afirst output channel of the second fusion gate configured to receive athird set of one or more qubits from the first input channel of thesecond fusion gate and the second input channel of the second fusiongate; a second output channel of the second fusion gate configured toreceive a fourth set of one or more qubits, distinct from the third setof one or more qubits, from the first input channel of the second fusiongate and the second input channel of the second fusion gate; a firstdetector of the second fusion gate configured to detect the third set ofone or more qubits; and a second detector of the second fusion gateconfigured to detect the second set of one or more qubits.
 12. Thedevice of claim 11, further including: a fourth beam splitter coupledwith: a first input channel for the fourth beam splitter configured forreceiving a first qubit of a qubit pair from a seventh qubit source ofthe plurality of qubit sources; and a second input channel for thefourth beam splitter configured for receiving a first qubit of a qubitpair from an eighth qubit source of the plurality of qubit sources; afirst output channel for the fourth beam splitter; and a second outputchannel for the fourth beam splitter; a third fusion gate coupled with:a first input channel, for the third fusion gate, coupled with thesecond output channel of the third beam splitter; and a second inputchannel, for the third fusion gate, coupled with the first outputchannel of the fourth beam splitter; and respective input channels forreceiving second qubits from: the qubit pair from the seventh qubitsource and the qubit pair from the eighth qubit source; wherein eachrespective input channel for receiving a second qubit is coupled with arespective output channel.
 13. The device of claim 12, wherein the thirdfusion gate includes: a first output channel of the third fusion gateconfigured to receive a fifth set of one or more qubits from the firstinput channel of the third fusion gate and the second input channel ofthe third fusion gate; a second output channel of the third fusion gateconfigured to receive a sixth set of one or more qubits, distinct fromthe fifth set of one or more qubits, from the first input channel of thethird fusion gate and the second input channel of the third fusion gate;a first detector of the third fusion gate configured to detect the thirdset of one or more qubits; and a second detector of the third fusiongate configured to detect the second set of one or more qubits.
 14. Thedevice of claim 12, wherein the third fusion gate includes: a firstoutput channel of the third fusion gate configured to receive a fifthset of one or more qubits from the first input channel of the thirdfusion gate and the second input channel of the third fusion gate; asecond output channel of the third fusion gate configured to receive asixth set of one or more qubits, distinct from the fifth set of one ormore qubits, from the first input channel of the third fusion gate andthe second input channel of the third fusion gate; and only one detectorof the third fusion gate configured to detect the third set of one ormore qubits; wherein the sixth set of one or more qubits is output fromthe third fusion gate along the second output channel.